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Estimation

linear-demand linear-cost, \(\sigma = 1\)

\(\theta = 0.05\)

\(\alpha_2 = 0.1\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.817 1.000 7.959 9.580 15.288
alpha1_hat 100 0 0.825 0.857 −1.470 0.766 3.413
alpha2_hat 100 0 0.110 0.081 −0.105 0.109 0.348
alpha3_hat 100 0 0.979 0.217 0.494 0.962 1.691
gamma0_hat 100 0 0.885 2.320 −17.279 1.197 4.363
gamma1_hat 100 0 2.497 9.899 −12.773 1.136 70.757
gamma2_hat 100 0 1.011 0.233 0.530 0.988 2.311
gamma3_hat 100 0 1.030 0.317 0.651 0.997 3.438
theta_hat 100 0 −0.808 5.535 −41.122 −0.060 9.001
theta_hat_t_stats 100 0 −0.107 1.107 −3.315 −0.044 2.587
R2_demand 100 0 0.370 0.292 −1.623 0.433 0.715
R2_supply 100 0 0.972 0.038 0.613 0.978 0.989
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.978 0.748 8.375 9.847 12.592
alpha1_hat 100 0 0.829 0.670 −0.830 0.745 2.707
alpha2_hat 100 0 0.116 0.061 −0.056 0.113 0.304
alpha3_hat 100 0 0.999 0.167 0.570 0.990 1.613
gamma0_hat 100 0 0.845 1.132 −2.634 0.791 2.968
gamma1_hat 100 0 0.865 1.887 −8.034 1.090 4.744
gamma2_hat 100 0 1.017 0.171 0.664 1.012 1.493
gamma3_hat 100 0 1.018 0.162 0.711 0.988 1.407
theta_hat 100 0 0.147 0.959 −2.028 0.059 4.702
theta_hat_t_stats 100 0 0.081 0.930 −1.825 0.123 2.907
R2_demand 100 0 0.325 0.241 −0.787 0.387 0.644
R2_supply 100 0 0.974 0.007 0.954 0.975 0.984
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.973 0.410 9.143 9.948 10.873
alpha1_hat 100 0 0.947 0.358 0.030 0.932 1.842
alpha2_hat 100 0 0.104 0.034 0.028 0.103 0.204
alpha3_hat 100 0 0.998 0.103 0.757 0.985 1.293
gamma0_hat 100 0 0.853 0.751 −0.897 0.999 2.610
gamma1_hat 100 0 1.016 0.984 −3.260 0.931 3.577
gamma2_hat 100 0 1.011 0.107 0.780 1.008 1.251
gamma3_hat 100 0 1.021 0.103 0.828 1.003 1.332
theta_hat 100 0 0.067 0.499 −1.557 0.102 2.117
theta_hat_t_stats 100 0 0.162 1.034 −2.557 0.317 2.408
R2_demand 100 0 0.332 0.127 −0.047 0.341 0.540
R2_supply 100 0 0.974 0.005 0.959 0.974 0.985
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.997 0.199 9.646 9.993 10.439
alpha1_hat 100 0 0.995 0.169 0.621 0.983 1.359
alpha2_hat 100 0 0.101 0.018 0.067 0.101 0.140
alpha3_hat 100 0 1.001 0.046 0.901 0.994 1.106
gamma0_hat 100 0 0.980 0.319 0.045 1.018 1.729
gamma1_hat 100 0 1.032 0.315 0.194 1.014 1.988
gamma2_hat 100 0 1.003 0.051 0.889 0.997 1.152
gamma3_hat 100 0 1.003 0.042 0.875 1.003 1.110
theta_hat 100 0 0.037 0.165 −0.484 0.047 0.487
theta_hat_t_stats 100 0 0.202 0.947 −2.282 0.293 2.470
R2_demand 100 0 0.342 0.050 0.237 0.340 0.458
R2_supply 100 0 0.974 0.002 0.969 0.974 0.979
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.996 0.146 9.640 9.993 10.428
alpha1_hat 100 0 1.012 0.118 0.683 1.016 1.274
alpha2_hat 100 0 0.099 0.011 0.073 0.099 0.124
alpha3_hat 100 0 1.000 0.037 0.909 1.002 1.081
gamma0_hat 100 0 1.037 0.203 0.572 1.049 1.469
gamma1_hat 100 0 0.997 0.244 0.376 1.000 1.534
gamma2_hat 100 0 0.996 0.031 0.926 0.994 1.078
gamma3_hat 100 0 1.000 0.026 0.931 1.000 1.060
theta_hat 100 0 0.046 0.130 −0.225 0.042 0.325
theta_hat_t_stats 100 0 0.384 1.082 −2.184 0.370 2.702
R2_demand 100 0 0.332 0.041 0.221 0.332 0.431
R2_supply 100 0 0.975 0.001 0.971 0.975 0.977
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.003 0.080 9.808 10.004 10.179
alpha1_hat 100 0 1.000 0.075 0.819 0.995 1.214
alpha2_hat 100 0 0.100 0.007 0.080 0.101 0.117
alpha3_hat 100 0 1.000 0.021 0.950 0.999 1.063
gamma0_hat 100 0 1.003 0.154 0.619 1.014 1.372
gamma1_hat 100 0 0.976 0.136 0.590 0.996 1.263
gamma2_hat 100 0 1.000 0.022 0.938 0.999 1.047
gamma3_hat 100 0 1.001 0.020 0.965 1.000 1.061
theta_hat 100 0 0.062 0.075 −0.088 0.051 0.284
theta_hat_t_stats 100 0 0.812 0.984 −1.240 0.688 3.691
R2_demand 100 0 0.336 0.022 0.287 0.336 0.389
R2_supply 100 0 0.974 0.001 0.972 0.974 0.977

\(\alpha_2 = 0.5\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.059 1.165 7.650 9.738 14.910
alpha1_hat 100 0 1.176 1.935 −3.982 1.278 7.462
alpha2_hat 100 0 0.490 0.207 0.124 0.465 1.534
alpha3_hat 100 0 0.985 0.246 0.477 0.942 1.737
gamma0_hat 100 0 0.729 1.769 −8.802 0.906 3.573
gamma1_hat 100 0 0.810 3.272 −5.914 0.868 15.210
gamma2_hat 100 0 1.029 0.261 0.555 1.012 2.072
gamma3_hat 100 0 1.049 0.268 0.410 0.999 2.476
theta_hat 100 0 0.129 0.535 −1.646 0.127 1.327
theta_hat_t_stats 100 0 0.167 1.045 −3.056 0.318 2.177
R2_demand 100 0 0.458 0.301 −1.023 0.537 0.783
R2_supply 100 0 0.964 0.016 0.857 0.967 0.989
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.954 0.651 8.636 9.896 11.753
alpha1_hat 100 0 1.072 1.230 −1.860 1.047 3.562
alpha2_hat 100 0 0.486 0.131 0.170 0.479 0.855
alpha3_hat 100 0 1.000 0.157 0.639 0.996 1.408
gamma0_hat 100 0 0.845 0.956 −1.477 0.926 2.493
gamma1_hat 100 0 0.945 1.557 −2.514 0.754 6.600
gamma2_hat 100 0 1.022 0.139 0.782 1.020 1.473
gamma3_hat 100 0 1.019 0.148 0.639 1.030 1.404
theta_hat 100 0 0.080 0.275 −1.013 0.106 0.552
theta_hat_t_stats 100 0 0.202 0.932 −2.695 0.400 2.030
R2_demand 100 0 0.513 0.142 0.085 0.535 0.744
R2_supply 100 0 0.964 0.008 0.941 0.966 0.980
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.044 0.566 8.881 10.032 11.537
alpha1_hat 100 0 1.174 1.117 −1.140 0.994 4.438
alpha2_hat 100 0 0.489 0.121 0.227 0.496 0.754
alpha3_hat 100 0 1.002 0.118 0.800 0.995 1.399
gamma0_hat 100 0 0.951 0.752 −1.084 0.967 2.826
gamma1_hat 100 0 1.059 1.226 −1.989 1.206 3.678
gamma2_hat 100 0 1.007 0.111 0.782 1.002 1.320
gamma3_hat 100 0 1.009 0.113 0.724 1.006 1.378
theta_hat 100 0 0.057 0.214 −0.394 0.017 0.707
theta_hat_t_stats 100 0 0.217 0.940 −2.144 0.093 2.636
R2_demand 100 0 0.510 0.116 0.133 0.530 0.706
R2_supply 100 0 0.965 0.006 0.950 0.965 0.979
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.016 0.175 9.584 10.020 10.482
alpha1_hat 100 0 1.003 0.360 0.058 0.995 1.761
alpha2_hat 100 0 0.501 0.041 0.418 0.497 0.596
alpha3_hat 100 0 1.004 0.044 0.928 1.001 1.117
gamma0_hat 100 0 0.993 0.262 0.376 0.990 1.768
gamma1_hat 100 0 1.031 0.453 0.014 1.033 2.079
gamma2_hat 100 0 0.999 0.041 0.880 1.001 1.117
gamma3_hat 100 0 0.996 0.043 0.879 0.996 1.117
theta_hat 100 0 0.046 0.085 −0.204 0.040 0.230
theta_hat_t_stats 100 0 0.519 0.985 −2.292 0.472 2.589
R2_demand 100 0 0.519 0.039 0.418 0.519 0.600
R2_supply 100 0 0.964 0.002 0.958 0.964 0.968
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.012 0.137 9.681 9.998 10.394
alpha1_hat 100 0 0.991 0.228 0.396 1.004 1.556
alpha2_hat 100 0 0.503 0.027 0.432 0.503 0.568
alpha3_hat 100 0 1.005 0.031 0.929 1.009 1.097
gamma0_hat 100 0 1.005 0.227 0.326 0.996 1.614
gamma1_hat 100 0 0.981 0.357 0.213 1.023 1.822
gamma2_hat 100 0 1.000 0.036 0.923 1.002 1.124
gamma3_hat 100 0 0.999 0.034 0.928 0.996 1.094
theta_hat 100 0 0.051 0.064 −0.082 0.047 0.204
theta_hat_t_stats 100 0 0.854 1.063 −1.409 0.789 3.143
R2_demand 100 0 0.520 0.031 0.432 0.520 0.580
R2_supply 100 0 0.964 0.002 0.959 0.964 0.969
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.008 0.089 9.823 10.002 10.274
alpha1_hat 100 0 1.009 0.192 0.503 1.023 1.425
alpha2_hat 100 0 0.500 0.019 0.454 0.501 0.557
alpha3_hat 100 0 0.999 0.019 0.947 1.001 1.035
gamma0_hat 100 0 0.984 0.119 0.739 0.983 1.287
gamma1_hat 100 0 1.014 0.190 0.295 1.013 1.393
gamma2_hat 100 0 1.001 0.018 0.959 1.002 1.042
gamma3_hat 100 0 1.003 0.020 0.958 1.002 1.046
theta_hat 100 0 0.050 0.036 −0.037 0.055 0.161
theta_hat_t_stats 100 0 1.340 0.947 −1.014 1.401 4.175
R2_demand 100 0 0.519 0.021 0.463 0.522 0.557
R2_supply 100 0 0.964 0.001 0.962 0.964 0.967

\(\alpha_2 = 1.0\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.933 0.897 7.453 9.922 12.191
alpha1_hat 100 0 0.943 3.335 −7.741 1.125 8.999
alpha2_hat 100 0 0.990 0.372 −0.355 0.966 1.926
alpha3_hat 100 0 0.992 0.218 0.563 0.996 1.477
gamma0_hat 100 0 0.640 2.061 −14.446 1.015 4.087
gamma1_hat 100 0 1.554 4.534 −18.420 1.443 16.969
gamma2_hat 100 0 1.059 0.302 0.575 1.000 3.281
gamma3_hat 100 0 1.069 0.261 0.610 1.043 2.256
theta_hat 100 0 0.052 0.626 −2.284 0.004 2.942
theta_hat_t_stats 100 0 −0.081 1.062 −2.654 0.020 1.924
R2_demand 100 0 0.599 0.163 −0.154 0.641 0.834
R2_supply 100 0 0.956 0.032 0.671 0.964 0.983
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.938 0.640 8.767 9.912 11.649
alpha1_hat 100 0 1.277 2.296 −3.572 1.307 9.090
alpha2_hat 100 0 0.957 0.243 0.171 0.962 1.500
alpha3_hat 100 0 0.996 0.141 0.591 0.976 1.348
gamma0_hat 100 0 0.750 1.261 −4.133 0.881 3.111
gamma1_hat 100 0 0.659 2.761 −10.647 0.663 7.674
gamma2_hat 100 0 1.029 0.177 0.625 1.014 1.746
gamma3_hat 100 0 1.012 0.174 0.624 0.990 1.609
theta_hat 100 0 0.116 0.303 −0.539 0.074 1.304
theta_hat_t_stats 100 0 0.282 0.945 −1.705 0.294 2.841
R2_demand 100 0 0.562 0.128 0.062 0.583 0.757
R2_supply 100 0 0.959 0.014 0.900 0.962 0.983
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.007 0.499 8.849 9.898 12.162
alpha1_hat 100 0 1.156 1.481 −2.597 1.140 5.041
alpha2_hat 100 0 0.986 0.184 0.567 0.987 1.489
alpha3_hat 100 0 0.997 0.102 0.805 0.980 1.371
gamma0_hat 100 0 0.951 0.702 −0.596 0.977 2.583
gamma1_hat 100 0 1.142 1.640 −3.068 1.222 5.098
gamma2_hat 100 0 1.012 0.100 0.806 1.008 1.282
gamma3_hat 100 0 1.010 0.102 0.796 1.011 1.316
theta_hat 100 0 0.039 0.159 −0.323 0.032 0.481
theta_hat_t_stats 100 0 0.207 0.903 −1.874 0.217 2.135
R2_demand 100 0 0.570 0.128 −0.289 0.581 0.734
R2_supply 100 0 0.960 0.007 0.938 0.961 0.977
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.006 0.190 9.664 10.007 10.476
alpha1_hat 100 0 1.074 0.713 −0.916 1.075 3.064
alpha2_hat 100 0 0.994 0.084 0.760 0.989 1.234
alpha3_hat 100 0 0.998 0.048 0.896 0.995 1.125
gamma0_hat 100 0 1.022 0.287 0.150 1.027 1.582
gamma1_hat 100 0 0.969 0.670 −1.149 1.006 2.646
gamma2_hat 100 0 0.996 0.043 0.874 0.992 1.111
gamma3_hat 100 0 1.009 0.040 0.916 1.009 1.120
theta_hat 100 0 0.052 0.072 −0.154 0.052 0.304
theta_hat_t_stats 100 0 0.666 0.959 −2.192 0.690 3.611
R2_demand 100 0 0.570 0.032 0.490 0.574 0.662
R2_supply 100 0 0.960 0.003 0.953 0.960 0.967
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.010 0.126 9.731 9.995 10.335
alpha1_hat 100 0 1.019 0.463 0.015 1.021 2.032
alpha2_hat 100 0 1.000 0.053 0.850 1.000 1.129
alpha3_hat 100 0 1.003 0.031 0.938 1.005 1.062
gamma0_hat 100 0 0.988 0.220 0.492 0.992 1.562
gamma1_hat 100 0 0.960 0.521 −0.133 0.954 2.213
gamma2_hat 100 0 1.003 0.032 0.919 1.006 1.068
gamma3_hat 100 0 0.998 0.029 0.913 0.995 1.085
theta_hat 100 0 0.054 0.049 −0.057 0.050 0.160
theta_hat_t_stats 100 0 1.018 0.926 −1.117 0.947 3.062
R2_demand 100 0 0.571 0.024 0.493 0.572 0.614
R2_supply 100 0 0.960 0.002 0.953 0.960 0.966
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.012 0.085 9.768 10.013 10.200
alpha1_hat 100 0 1.054 0.308 0.219 1.050 1.716
alpha2_hat 100 0 0.997 0.033 0.930 0.993 1.075
alpha3_hat 100 0 1.003 0.020 0.959 1.002 1.049
gamma0_hat 100 0 1.002 0.144 0.629 1.009 1.361
gamma1_hat 100 0 0.988 0.338 0.080 1.025 1.926
gamma2_hat 100 0 1.001 0.020 0.954 1.001 1.053
gamma3_hat 100 0 0.999 0.022 0.933 1.000 1.051
theta_hat 100 0 0.050 0.035 −0.040 0.051 0.141
theta_hat_t_stats 100 0 1.521 1.030 −1.231 1.583 4.201
R2_demand 100 0 0.567 0.017 0.533 0.569 0.607
R2_supply 100 0 0.960 0.002 0.956 0.960 0.963

\(\alpha_2 = 5.0\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.027 0.973 8.184 9.967 14.140
alpha1_hat 100 0 2.793 15.115 −34.879 2.222 36.412
alpha2_hat 100 0 4.848 1.764 1.841 4.817 9.933
alpha3_hat 100 0 1.004 0.222 0.481 0.970 1.743
gamma0_hat 100 0 0.904 1.700 −4.256 1.140 3.702
gamma1_hat 100 0 3.342 16.518 −32.573 3.749 46.579
gamma2_hat 100 0 0.987 0.221 0.530 0.970 1.602
gamma3_hat 100 0 1.025 0.255 0.477 1.000 2.254
theta_hat 100 0 0.035 0.415 −0.927 −0.009 1.357
theta_hat_t_stats 100 0 −0.062 0.940 −1.957 −0.037 1.841
R2_demand 100 0 0.591 0.189 −0.415 0.620 0.827
R2_supply 100 0 0.951 0.029 0.791 0.960 0.983
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.961 0.545 8.679 9.945 11.240
alpha1_hat 100 0 1.358 8.303 −18.125 1.695 18.536
alpha2_hat 100 0 4.939 0.965 2.676 4.963 7.482
alpha3_hat 100 0 0.994 0.134 0.628 0.996 1.392
gamma0_hat 100 0 0.870 0.891 −2.312 0.924 2.966
gamma1_hat 100 0 −1.595 10.364 −32.843 −1.492 21.853
gamma2_hat 100 0 1.018 0.133 0.716 1.011 1.456
gamma3_hat 100 0 1.023 0.151 0.691 1.036 1.395
theta_hat 100 0 0.121 0.227 −0.359 0.120 0.947
theta_hat_t_stats 100 0 0.443 0.912 −2.061 0.546 2.856
R2_demand 100 0 0.622 0.105 0.291 0.640 0.816
R2_supply 100 0 0.955 0.012 0.910 0.957 0.975
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.967 0.396 9.046 9.941 10.802
alpha1_hat 100 0 0.952 7.211 −15.549 1.057 18.070
alpha2_hat 100 0 4.972 0.807 3.013 4.915 6.846
alpha3_hat 100 0 1.001 0.099 0.769 0.993 1.269
gamma0_hat 100 0 1.000 0.728 −0.759 1.040 2.401
gamma1_hat 100 0 1.201 7.393 −16.017 0.730 22.576
gamma2_hat 100 0 0.997 0.105 0.778 0.991 1.339
gamma3_hat 100 0 1.008 0.109 0.781 0.998 1.265
theta_hat 100 0 0.052 0.179 −0.443 0.026 0.470
theta_hat_t_stats 100 0 0.214 1.078 −2.563 0.218 2.362
R2_demand 100 0 0.620 0.068 0.439 0.633 0.746
R2_supply 100 0 0.955 0.009 0.922 0.956 0.970
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.021 0.196 9.574 9.984 10.626
alpha1_hat 100 0 1.292 2.799 −5.567 0.953 9.036
alpha2_hat 100 0 4.988 0.341 4.140 4.967 5.810
alpha3_hat 100 0 1.001 0.047 0.903 1.002 1.120
gamma0_hat 100 0 1.023 0.296 0.171 1.025 1.626
gamma1_hat 100 0 0.913 3.009 −5.073 1.151 8.242
gamma2_hat 100 0 0.996 0.043 0.900 0.994 1.111
gamma3_hat 100 0 1.002 0.046 0.881 0.998 1.107
theta_hat 100 0 0.049 0.069 −0.114 0.044 0.193
theta_hat_t_stats 100 0 0.717 1.039 −1.909 0.685 2.878
R2_demand 100 0 0.617 0.036 0.502 0.619 0.691
R2_supply 100 0 0.955 0.003 0.947 0.955 0.961
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.003 0.139 9.664 9.999 10.391
alpha1_hat 100 0 0.892 1.788 −4.114 0.834 4.986
alpha2_hat 100 0 5.011 0.207 4.459 5.035 5.508
alpha3_hat 100 0 0.999 0.029 0.933 0.997 1.060
gamma0_hat 100 0 1.027 0.191 0.554 1.038 1.529
gamma1_hat 100 0 0.824 2.086 −5.515 0.945 6.108
gamma2_hat 100 0 0.998 0.028 0.927 0.999 1.073
gamma3_hat 100 0 0.997 0.032 0.924 0.995 1.068
theta_hat 100 0 0.050 0.046 −0.063 0.052 0.192
theta_hat_t_stats 100 0 1.057 0.982 −1.450 1.124 3.838
R2_demand 100 0 0.619 0.023 0.550 0.624 0.662
R2_supply 100 0 0.955 0.002 0.948 0.955 0.961
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.000 0.082 9.776 9.995 10.227
alpha1_hat 100 0 0.978 1.209 −2.241 1.031 4.140
alpha2_hat 100 0 5.003 0.142 4.679 4.974 5.388
alpha3_hat 100 0 1.001 0.019 0.959 1.000 1.047
gamma0_hat 100 0 1.013 0.119 0.726 1.010 1.293
gamma1_hat 100 0 0.783 1.292 −2.388 0.935 3.540
gamma2_hat 100 0 1.000 0.018 0.954 1.000 1.049
gamma3_hat 100 0 1.000 0.019 0.927 1.001 1.047
theta_hat 100 0 0.052 0.026 −0.005 0.049 0.119
theta_hat_t_stats 100 0 1.788 0.853 −0.168 1.738 3.926
R2_demand 100 0 0.620 0.014 0.587 0.621 0.653
R2_supply 100 0 0.955 0.001 0.950 0.955 0.958

\(\alpha_2 = 20.0\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.078 1.156 7.657 10.040 17.632
alpha1_hat 100 0 2.394 47.415 −116.140 1.858 116.686
alpha2_hat 100 0 20.067 5.963 8.920 19.537 49.105
alpha3_hat 100 0 1.028 0.243 0.532 0.999 2.502
gamma0_hat 100 0 1.099 1.335 −5.080 1.365 3.467
gamma1_hat 100 0 11.897 57.608 −94.735 18.775 208.804
gamma2_hat 100 0 0.987 0.216 0.494 0.972 1.955
gamma3_hat 100 0 0.983 0.224 0.552 0.945 1.687
theta_hat 100 0 −0.012 0.361 −1.222 −0.025 1.432
theta_hat_t_stats 100 0 −0.133 0.987 −2.743 −0.090 1.916
R2_demand 100 0 0.564 0.537 −4.527 0.655 0.827
R2_supply 100 0 0.955 0.019 0.883 0.959 0.978
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.000 0.682 8.803 9.929 12.743
alpha1_hat 100 0 2.544 33.319 −74.863 0.205 79.604
alpha2_hat 100 0 19.850 3.830 11.814 20.057 30.398
alpha3_hat 100 0 0.992 0.157 0.692 0.984 1.611
gamma0_hat 100 0 0.911 0.894 −1.489 0.976 2.832
gamma1_hat 100 0 −0.519 40.158 −115.921 −0.394 114.922
gamma2_hat 100 0 1.021 0.138 0.680 1.028 1.385
gamma3_hat 100 0 1.006 0.140 0.695 0.998 1.461
theta_hat 100 0 0.067 0.216 −0.399 0.048 0.700
theta_hat_t_stats 100 0 0.207 0.973 −2.418 0.258 2.427
R2_demand 100 0 0.618 0.129 −0.022 0.636 0.788
R2_supply 100 0 0.954 0.013 0.912 0.957 0.975
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.994 0.371 9.163 10.035 11.065
alpha1_hat 100 0 0.532 21.853 −67.392 −0.577 48.025
alpha2_hat 100 0 20.039 2.642 13.664 20.145 27.673
alpha3_hat 100 0 0.991 0.103 0.703 1.002 1.231
gamma0_hat 100 0 0.994 0.802 −1.146 1.086 2.491
gamma1_hat 100 0 3.179 21.209 −59.636 3.509 66.693
gamma2_hat 100 0 1.008 0.110 0.778 0.990 1.279
gamma3_hat 100 0 1.004 0.107 0.793 0.991 1.428
theta_hat 100 0 0.037 0.144 −0.300 0.038 0.415
theta_hat_t_stats 100 0 0.170 0.935 −2.326 0.258 2.641
R2_demand 100 0 0.633 0.073 0.391 0.644 0.773
R2_supply 100 0 0.954 0.009 0.919 0.955 0.971
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.019 0.174 9.616 10.026 10.367
alpha1_hat 100 0 −0.831 10.322 −28.642 −1.953 26.857
alpha2_hat 100 0 20.251 1.177 16.694 20.211 23.084
alpha3_hat 100 0 1.006 0.041 0.912 1.012 1.103
gamma0_hat 100 0 0.964 0.300 0.199 0.960 1.919
gamma1_hat 100 0 −1.540 10.888 −31.544 −2.235 25.781
gamma2_hat 100 0 1.008 0.043 0.883 1.004 1.144
gamma3_hat 100 0 1.001 0.045 0.859 1.001 1.124
theta_hat 100 0 0.065 0.064 −0.073 0.064 0.204
theta_hat_t_stats 100 0 0.983 0.966 −1.289 0.993 3.082
R2_demand 100 0 0.634 0.028 0.545 0.635 0.696
R2_supply 100 0 0.953 0.004 0.940 0.953 0.962
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.999 0.137 9.610 10.001 10.327
alpha1_hat 100 0 1.356 8.204 −15.540 0.259 21.595
alpha2_hat 100 0 19.974 0.969 18.050 19.986 22.431
alpha3_hat 100 0 1.004 0.032 0.889 1.004 1.073
gamma0_hat 100 0 1.013 0.197 0.398 1.013 1.388
gamma1_hat 100 0 1.539 7.142 −15.781 1.533 22.144
gamma2_hat 100 0 1.000 0.031 0.932 0.999 1.092
gamma3_hat 100 0 0.997 0.033 0.912 0.993 1.077
theta_hat 100 0 0.045 0.041 −0.050 0.047 0.141
theta_hat_t_stats 100 0 0.983 0.887 −1.082 1.037 3.040
R2_demand 100 0 0.630 0.022 0.580 0.629 0.688
R2_supply 100 0 0.953 0.003 0.945 0.954 0.959
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.991 0.076 9.851 9.981 10.176
alpha1_hat 100 0 0.919 4.200 −9.902 0.690 14.379
alpha2_hat 100 0 19.977 0.462 18.729 20.022 21.168
alpha3_hat 100 0 1.000 0.017 0.952 1.000 1.040
gamma0_hat 100 0 1.009 0.132 0.632 1.017 1.325
gamma1_hat 100 0 1.182 4.781 −9.824 0.773 13.812
gamma2_hat 100 0 1.000 0.021 0.946 0.998 1.061
gamma3_hat 100 0 1.000 0.019 0.953 1.001 1.050
theta_hat 100 0 0.048 0.027 −0.040 0.048 0.112
theta_hat_t_stats 100 0 1.654 0.931 −1.431 1.700 3.756
R2_demand 100 0 0.631 0.013 0.602 0.633 0.659
R2_supply 100 0 0.953 0.001 0.949 0.953 0.957

\(\theta = 0.1\)

\(\alpha_2 = 0.1\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.922 1.004 7.786 9.863 14.467
alpha1_hat 100 0 0.980 0.866 −0.802 0.867 3.118
alpha2_hat 100 0 0.097 0.084 −0.101 0.096 0.362
alpha3_hat 100 0 0.993 0.247 0.475 0.952 2.143
gamma0_hat 100 0 0.696 1.897 −6.221 1.122 4.230
gamma1_hat 100 0 0.268 6.858 −39.767 0.995 20.044
gamma2_hat 100 0 1.029 0.259 0.404 0.976 1.805
gamma3_hat 100 0 1.045 0.267 0.450 1.008 1.922
theta_hat 100 0 0.585 3.766 −8.145 0.146 22.392
theta_hat_t_stats 100 0 0.144 1.034 −2.194 0.184 3.212
R2_demand 100 0 0.313 0.320 −1.342 0.399 0.705
R2_supply 100 0 0.974 0.016 0.862 0.977 0.990
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.982 0.793 8.536 9.913 14.488
alpha1_hat 100 0 0.983 0.706 −0.476 0.892 4.420
alpha2_hat 100 0 0.101 0.061 −0.055 0.101 0.364
alpha3_hat 100 0 0.998 0.183 0.695 0.974 2.060
gamma0_hat 100 0 0.951 0.981 −3.125 0.974 3.052
gamma1_hat 100 0 1.664 7.218 −18.935 1.102 56.585
gamma2_hat 100 0 1.005 0.144 0.694 1.005 1.527
gamma3_hat 100 0 1.008 0.161 0.585 0.980 1.368
theta_hat 100 0 0.002 2.669 −15.902 0.117 11.026
theta_hat_t_stats 100 0 0.215 1.044 −2.261 0.120 2.577
R2_demand 100 0 0.279 0.311 −2.128 0.345 0.582
R2_supply 100 0 0.975 0.007 0.936 0.976 0.987
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.017 0.514 8.751 9.988 11.461
alpha1_hat 100 0 1.064 0.383 0.235 1.066 1.977
alpha2_hat 100 0 0.094 0.040 0.005 0.091 0.220
alpha3_hat 100 0 1.007 0.101 0.823 0.997 1.310
gamma0_hat 100 0 0.861 0.777 −1.152 0.850 2.406
gamma1_hat 100 0 0.916 1.588 −10.603 1.041 5.889
gamma2_hat 100 0 1.013 0.111 0.809 1.003 1.346
gamma3_hat 100 0 1.023 0.114 0.790 1.024 1.432
theta_hat 100 0 0.160 0.803 −2.355 0.113 5.796
theta_hat_t_stats 100 0 0.265 1.075 −2.432 0.237 3.224
R2_demand 100 0 0.310 0.134 −0.144 0.327 0.574
R2_supply 100 0 0.974 0.005 0.960 0.975 0.984
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.008 0.185 9.641 10.009 10.470
alpha1_hat 100 0 0.987 0.168 0.606 0.985 1.521
alpha2_hat 100 0 0.102 0.016 0.055 0.102 0.131
alpha3_hat 100 0 0.998 0.046 0.873 0.998 1.120
gamma0_hat 100 0 0.991 0.326 0.062 1.004 1.792
gamma1_hat 100 0 1.052 0.348 0.132 1.088 1.913
gamma2_hat 100 0 0.999 0.044 0.899 0.999 1.099
gamma3_hat 100 0 1.002 0.051 0.881 1.003 1.119
theta_hat 100 0 0.076 0.187 −0.354 0.068 0.628
theta_hat_t_stats 100 0 0.412 1.084 −2.290 0.453 3.714
R2_demand 100 0 0.313 0.051 0.186 0.323 0.403
R2_supply 100 0 0.975 0.002 0.970 0.976 0.980
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.998 0.150 9.650 10.002 10.424
alpha1_hat 100 0 0.987 0.123 0.569 0.993 1.312
alpha2_hat 100 0 0.101 0.012 0.072 0.101 0.125
alpha3_hat 100 0 1.001 0.036 0.910 1.001 1.110
gamma0_hat 100 0 0.970 0.226 0.345 0.960 1.502
gamma1_hat 100 0 0.992 0.252 0.176 1.006 1.554
gamma2_hat 100 0 1.004 0.034 0.920 1.001 1.104
gamma3_hat 100 0 1.004 0.032 0.922 1.008 1.082
theta_hat 100 0 0.110 0.131 −0.196 0.093 0.498
theta_hat_t_stats 100 0 0.872 1.015 −1.513 0.796 3.817
R2_demand 100 0 0.312 0.040 0.198 0.314 0.415
R2_supply 100 0 0.975 0.001 0.972 0.975 0.979
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.007 0.082 9.812 10.004 10.206
alpha1_hat 100 0 0.989 0.081 0.798 0.978 1.193
alpha2_hat 100 0 0.102 0.008 0.084 0.101 0.120
alpha3_hat 100 0 1.002 0.020 0.962 1.003 1.054
gamma0_hat 100 0 0.973 0.133 0.567 0.976 1.274
gamma1_hat 100 0 1.015 0.138 0.556 1.023 1.456
gamma2_hat 100 0 1.004 0.020 0.949 1.002 1.050
gamma3_hat 100 0 1.002 0.019 0.948 1.003 1.051
theta_hat 100 0 0.096 0.071 −0.098 0.095 0.329
theta_hat_t_stats 100 0 1.269 0.923 −1.143 1.203 4.404
R2_demand 100 0 0.312 0.026 0.243 0.314 0.360
R2_supply 100 0 0.975 0.001 0.973 0.975 0.978

\(\alpha_2 = 0.5\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.978 1.100 7.673 9.744 14.551
alpha1_hat 100 0 1.068 2.043 −3.604 1.104 9.409
alpha2_hat 100 0 0.491 0.222 −0.065 0.470 1.043
alpha3_hat 100 0 0.979 0.220 0.433 0.949 1.703
gamma0_hat 100 0 0.801 1.611 −3.806 1.089 4.233
gamma1_hat 100 0 38.189 374.984 −31.951 1.125 3750.284
gamma2_hat 100 0 1.021 0.222 0.652 0.990 1.638
gamma3_hat 100 0 1.023 0.223 0.481 0.998 1.602
theta_hat 100 0 −8.140 83.354 −833.306 0.038 6.903
theta_hat_t_stats 100 0 0.146 0.938 −2.223 0.108 2.629
R2_demand 100 0 0.441 0.308 −1.486 0.501 0.798
R2_supply 100 0 0.965 0.015 0.890 0.969 0.988
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.938 0.726 8.691 9.844 12.232
alpha1_hat 100 0 1.088 1.283 −2.326 1.066 5.789
alpha2_hat 100 0 0.484 0.140 0.028 0.491 0.981
alpha3_hat 100 0 0.991 0.157 0.605 0.962 1.598
gamma0_hat 100 0 1.110 0.972 −1.632 1.195 3.511
gamma1_hat 100 0 0.516 4.430 −39.831 0.874 7.488
gamma2_hat 100 0 0.976 0.154 0.636 0.955 1.475
gamma3_hat 100 0 0.980 0.146 0.613 0.971 1.341
theta_hat 100 0 0.191 0.917 −0.942 0.095 8.588
theta_hat_t_stats 100 0 0.275 1.017 −2.654 0.406 3.067
R2_demand 100 0 0.469 0.192 −0.342 0.511 0.774
R2_supply 100 0 0.968 0.009 0.928 0.968 0.982
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.041 0.475 9.076 10.000 11.728
alpha1_hat 100 0 1.054 1.037 −1.468 0.926 3.057
alpha2_hat 100 0 0.500 0.112 0.204 0.497 0.780
alpha3_hat 100 0 1.010 0.110 0.765 1.016 1.246
gamma0_hat 100 0 0.874 0.770 −1.888 0.993 2.677
gamma1_hat 100 0 1.112 1.128 −1.090 1.063 4.240
gamma2_hat 100 0 1.019 0.116 0.767 1.014 1.350
gamma3_hat 100 0 0.993 0.102 0.770 0.980 1.389
theta_hat 100 0 0.094 0.214 −0.393 0.088 0.615
theta_hat_t_stats 100 0 0.403 1.029 −2.875 0.385 2.805
R2_demand 100 0 0.477 0.126 −0.315 0.500 0.643
R2_supply 100 0 0.965 0.007 0.939 0.966 0.978
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.042 0.185 9.652 10.034 10.520
alpha1_hat 100 0 0.969 0.399 −0.250 0.997 2.022
alpha2_hat 100 0 0.509 0.044 0.410 0.505 0.621
alpha3_hat 100 0 1.014 0.047 0.912 1.011 1.141
gamma0_hat 100 0 1.023 0.258 0.411 1.023 1.605
gamma1_hat 100 0 0.937 0.426 −0.057 0.947 1.992
gamma2_hat 100 0 0.996 0.041 0.920 0.992 1.102
gamma3_hat 100 0 1.002 0.040 0.920 1.002 1.118
theta_hat 100 0 0.108 0.074 −0.061 0.110 0.270
theta_hat_t_stats 100 0 1.252 0.847 −0.708 1.309 3.174
R2_demand 100 0 0.480 0.045 0.378 0.482 0.572
R2_supply 100 0 0.967 0.002 0.960 0.967 0.972
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.972 0.131 9.681 9.966 10.273
alpha1_hat 100 0 0.966 0.276 −0.048 0.946 1.588
alpha2_hat 100 0 0.500 0.029 0.443 0.499 0.601
alpha3_hat 100 0 0.997 0.031 0.923 0.993 1.096
gamma0_hat 100 0 1.012 0.207 0.396 1.033 1.514
gamma1_hat 100 0 0.965 0.351 0.208 0.947 1.764
gamma2_hat 100 0 0.999 0.028 0.933 0.997 1.081
gamma3_hat 100 0 1.000 0.030 0.938 0.997 1.084
theta_hat 100 0 0.105 0.062 −0.028 0.097 0.262
theta_hat_t_stats 100 0 1.705 0.994 −0.422 1.604 3.981
R2_demand 100 0 0.494 0.031 0.424 0.492 0.594
R2_supply 100 0 0.967 0.002 0.960 0.967 0.971
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.005 0.087 9.848 9.996 10.264
alpha1_hat 100 0 1.028 0.182 0.705 1.010 1.504
alpha2_hat 100 0 0.498 0.019 0.447 0.501 0.534
alpha3_hat 100 0 0.999 0.019 0.961 0.995 1.057
gamma0_hat 100 0 1.031 0.122 0.698 1.035 1.301
gamma1_hat 100 0 0.997 0.223 0.501 0.996 1.425
gamma2_hat 100 0 0.996 0.019 0.956 0.998 1.055
gamma3_hat 100 0 0.998 0.019 0.952 0.998 1.047
theta_hat 100 0 0.097 0.039 0.024 0.100 0.191
theta_hat_t_stats 100 0 2.475 0.959 0.581 2.609 4.749
R2_demand 100 0 0.490 0.020 0.428 0.492 0.528
R2_supply 100 0 0.966 0.001 0.964 0.966 0.970

\(\alpha_2 = 1.0\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.038 0.939 8.563 9.859 13.464
alpha1_hat 100 0 1.139 3.316 −7.905 0.999 9.244
alpha2_hat 100 0 0.998 0.364 0.259 0.969 1.958
alpha3_hat 100 0 1.000 0.203 0.591 0.988 1.603
gamma0_hat 100 0 0.376 2.091 −14.550 0.795 3.760
gamma1_hat 100 0 1.860 5.439 −14.769 2.191 24.736
gamma2_hat 100 0 1.071 0.292 0.610 1.017 2.916
gamma3_hat 100 0 1.070 0.310 0.377 1.040 3.055
theta_hat 100 0 0.137 0.584 −1.691 0.056 2.587
theta_hat_t_stats 100 0 0.085 0.963 −3.160 0.183 2.192
R2_demand 100 0 0.501 0.210 −0.175 0.550 0.766
R2_supply 100 0 0.958 0.035 0.653 0.965 0.982
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.998 0.699 8.589 10.008 12.681
alpha1_hat 100 0 0.966 2.437 −7.025 0.897 6.588
alpha2_hat 100 0 1.004 0.283 0.358 1.009 1.796
alpha3_hat 100 0 0.992 0.162 0.666 0.982 1.479
gamma0_hat 100 0 0.748 0.990 −1.939 0.766 3.205
gamma1_hat 100 0 0.915 3.725 −13.519 1.156 11.311
gamma2_hat 100 0 1.033 0.153 0.603 1.022 1.462
gamma3_hat 100 0 1.025 0.130 0.778 1.029 1.371
theta_hat 100 0 0.145 0.384 −0.731 0.104 1.793
theta_hat_t_stats 100 0 0.380 1.064 −2.135 0.468 2.402
R2_demand 100 0 0.535 0.142 0.009 0.559 0.772
R2_supply 100 0 0.961 0.010 0.921 0.963 0.979
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.052 0.470 9.294 9.975 11.816
alpha1_hat 100 0 0.985 1.597 −3.600 1.158 5.063
alpha2_hat 100 0 1.013 0.187 0.547 0.993 1.675
alpha3_hat 100 0 1.009 0.109 0.786 1.004 1.354
gamma0_hat 100 0 0.961 0.587 −1.124 0.952 2.288
gamma1_hat 100 0 0.993 1.830 −3.805 0.966 5.987
gamma2_hat 100 0 1.006 0.091 0.814 0.997 1.297
gamma3_hat 100 0 1.002 0.093 0.788 1.008 1.356
theta_hat 100 0 0.102 0.176 −0.346 0.109 0.671
theta_hat_t_stats 100 0 0.548 0.931 −1.758 0.557 2.690
R2_demand 100 0 0.529 0.112 −0.065 0.552 0.714
R2_supply 100 0 0.962 0.007 0.937 0.963 0.975
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.016 0.198 9.512 9.995 10.631
alpha1_hat 100 0 1.035 0.685 −1.053 1.009 3.025
alpha2_hat 100 0 0.999 0.082 0.797 1.003 1.248
alpha3_hat 100 0 0.997 0.049 0.896 0.992 1.156
gamma0_hat 100 0 0.962 0.344 0.164 0.957 1.892
gamma1_hat 100 0 1.096 0.806 −0.830 1.103 2.837
gamma2_hat 100 0 1.004 0.050 0.886 1.007 1.116
gamma3_hat 100 0 0.997 0.048 0.864 0.997 1.141
theta_hat 100 0 0.097 0.084 −0.096 0.100 0.272
theta_hat_t_stats 100 0 1.216 1.050 −1.298 1.289 3.237
R2_demand 100 0 0.529 0.041 0.420 0.531 0.624
R2_supply 100 0 0.962 0.003 0.954 0.962 0.969
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.994 0.127 9.689 9.981 10.334
alpha1_hat 100 0 1.076 0.462 0.143 1.053 2.293
alpha2_hat 100 0 0.991 0.046 0.883 0.987 1.095
alpha3_hat 100 0 1.002 0.031 0.902 1.001 1.075
gamma0_hat 100 0 1.013 0.208 0.495 1.007 1.482
gamma1_hat 100 0 0.957 0.511 −0.863 0.995 2.257
gamma2_hat 100 0 1.000 0.029 0.922 1.001 1.081
gamma3_hat 100 0 0.996 0.033 0.923 0.996 1.069
theta_hat 100 0 0.101 0.052 −0.046 0.094 0.284
theta_hat_t_stats 100 0 1.840 0.920 −0.871 1.732 4.718
R2_demand 100 0 0.540 0.023 0.480 0.543 0.612
R2_supply 100 0 0.963 0.002 0.957 0.963 0.968
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.002 0.094 9.754 10.008 10.219
alpha1_hat 100 0 0.994 0.269 0.298 0.992 1.531
alpha2_hat 100 0 1.001 0.032 0.908 0.999 1.076
alpha3_hat 100 0 1.000 0.021 0.955 0.997 1.054
gamma0_hat 100 0 1.011 0.116 0.727 1.013 1.305
gamma1_hat 100 0 1.011 0.271 0.430 1.011 1.535
gamma2_hat 100 0 0.998 0.019 0.951 0.998 1.039
gamma3_hat 100 0 0.996 0.020 0.949 1.000 1.039
theta_hat 100 0 0.098 0.028 0.036 0.099 0.158
theta_hat_t_stats 100 0 2.870 0.771 1.113 2.894 4.520
R2_demand 100 0 0.539 0.018 0.494 0.538 0.586
R2_supply 100 0 0.963 0.001 0.959 0.963 0.966

\(\alpha_2 = 5.0\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.972 0.721 8.034 9.826 12.227
alpha1_hat 100 0 0.732 12.626 −30.189 1.279 31.383
alpha2_hat 100 0 5.012 1.345 2.416 4.866 8.509
alpha3_hat 100 0 1.007 0.213 0.575 0.995 1.646
gamma0_hat 100 0 0.651 2.007 −7.080 1.176 3.719
gamma1_hat 100 0 0.163 15.866 −44.945 0.729 35.737
gamma2_hat 100 0 1.039 0.290 0.616 0.990 2.241
gamma3_hat 100 0 1.038 0.251 0.606 1.001 1.983
theta_hat 100 0 0.150 0.404 −0.480 0.078 1.966
theta_hat_t_stats 100 0 0.231 0.916 −1.868 0.318 1.936
R2_demand 100 0 0.563 0.203 −0.275 0.602 0.819
R2_supply 100 0 0.956 0.024 0.828 0.963 0.985
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.938 0.567 8.736 9.990 11.303
alpha1_hat 100 0 1.609 10.753 −29.624 1.950 38.203
alpha2_hat 100 0 4.877 1.207 1.285 4.847 9.029
alpha3_hat 100 0 1.003 0.133 0.770 0.995 1.363
gamma0_hat 100 0 0.889 0.935 −2.240 0.995 3.073
gamma1_hat 100 0 −0.993 11.173 −57.032 −0.681 25.385
gamma2_hat 100 0 1.017 0.137 0.690 0.991 1.433
gamma3_hat 100 0 1.031 0.155 0.659 1.015 1.486
theta_hat 100 0 0.154 0.248 −0.318 0.129 1.411
theta_hat_t_stats 100 0 0.523 0.771 −1.730 0.541 2.136
R2_demand 100 0 0.591 0.112 0.243 0.613 0.799
R2_supply 100 0 0.958 0.011 0.925 0.961 0.977
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.030 0.447 8.912 9.959 11.050
alpha1_hat 100 0 0.800 6.976 −16.810 0.746 18.238
alpha2_hat 100 0 5.048 0.844 3.433 5.046 7.572
alpha3_hat 100 0 1.011 0.106 0.760 1.009 1.282
gamma0_hat 100 0 0.899 0.652 −0.767 0.956 2.249
gamma1_hat 100 0 1.289 7.650 −19.914 1.022 22.499
gamma2_hat 100 0 1.015 0.102 0.838 0.999 1.327
gamma3_hat 100 0 1.016 0.101 0.780 1.010 1.279
theta_hat 100 0 0.107 0.165 −0.308 0.107 0.583
theta_hat_t_stats 100 0 0.615 0.968 −2.240 0.691 2.904
R2_demand 100 0 0.576 0.088 0.266 0.592 0.736
R2_supply 100 0 0.958 0.007 0.941 0.959 0.972
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.000 0.183 9.653 9.977 10.385
alpha1_hat 100 0 1.291 2.956 −5.136 1.479 7.205
alpha2_hat 100 0 4.973 0.360 4.186 4.986 5.969
alpha3_hat 100 0 1.008 0.041 0.921 1.010 1.142
gamma0_hat 100 0 0.934 0.307 0.133 0.930 1.603
gamma1_hat 100 0 1.040 2.853 −6.274 0.913 6.886
gamma2_hat 100 0 1.008 0.047 0.881 1.004 1.116
gamma3_hat 100 0 1.008 0.041 0.923 1.013 1.109
theta_hat 100 0 0.107 0.067 −0.067 0.112 0.262
theta_hat_t_stats 100 0 1.503 0.905 −1.128 1.638 3.560
R2_demand 100 0 0.588 0.034 0.509 0.589 0.674
R2_supply 100 0 0.958 0.004 0.944 0.958 0.965
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.985 0.133 9.706 9.966 10.504
alpha1_hat 100 0 0.858 2.075 −4.874 0.888 6.281
alpha2_hat 100 0 4.998 0.252 4.389 4.956 5.689
alpha3_hat 100 0 0.998 0.028 0.949 0.999 1.086
gamma0_hat 100 0 1.003 0.210 0.418 0.995 1.624
gamma1_hat 100 0 0.824 2.211 −4.046 0.848 6.384
gamma2_hat 100 0 0.999 0.030 0.925 1.001 1.076
gamma3_hat 100 0 1.003 0.038 0.910 1.004 1.096
theta_hat 100 0 0.104 0.054 −0.024 0.105 0.218
theta_hat_t_stats 100 0 2.106 1.054 −0.536 2.193 4.360
R2_demand 100 0 0.589 0.025 0.499 0.591 0.629
R2_supply 100 0 0.958 0.002 0.952 0.958 0.964
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.999 0.086 9.818 9.988 10.300
alpha1_hat 100 0 0.973 1.200 −1.703 0.997 3.804
alpha2_hat 100 0 5.003 0.145 4.683 5.007 5.338
alpha3_hat 100 0 1.001 0.020 0.954 0.999 1.047
gamma0_hat 100 0 1.013 0.133 0.754 1.013 1.305
gamma1_hat 100 0 1.150 1.308 −1.514 1.238 4.211
gamma2_hat 100 0 0.999 0.020 0.956 0.996 1.046
gamma3_hat 100 0 0.997 0.018 0.947 0.996 1.046
theta_hat 100 0 0.095 0.029 0.030 0.098 0.161
theta_hat_t_stats 100 0 3.106 0.905 0.993 3.163 5.110
R2_demand 100 0 0.586 0.019 0.531 0.586 0.626
R2_supply 100 0 0.958 0.001 0.955 0.958 0.962

\(\alpha_2 = 20.0\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.009 0.849 8.005 9.985 12.852
alpha1_hat 100 0 −1.334 57.303 −166.001 −4.268 170.300
alpha2_hat 100 0 20.221 6.912 0.669 19.854 41.702
alpha3_hat 100 0 0.983 0.197 0.505 0.978 1.448
gamma0_hat 100 0 0.537 2.043 −8.719 0.972 3.459
gamma1_hat 100 0 −24.302 161.538 −1481.399 −2.022 182.283
gamma2_hat 100 0 1.057 0.237 0.517 1.042 1.888
gamma3_hat 100 0 1.070 0.303 0.590 1.021 2.912
theta_hat 100 0 0.311 0.970 −0.520 0.131 8.392
theta_hat_t_stats 100 0 0.371 0.953 −2.341 0.449 2.521
R2_demand 100 0 0.585 0.161 −0.060 0.619 0.858
R2_supply 100 0 0.954 0.038 0.738 0.964 0.982
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.136 0.644 8.713 10.140 11.647
alpha1_hat 100 0 −0.463 41.128 −102.521 0.213 130.323
alpha2_hat 100 0 20.640 4.822 8.791 20.220 31.189
alpha3_hat 100 0 1.009 0.145 0.699 1.004 1.313
gamma0_hat 100 0 0.813 1.015 −2.678 0.853 2.763
gamma1_hat 100 0 4.002 41.292 −96.370 7.400 91.148
gamma2_hat 100 0 1.025 0.163 0.713 1.016 1.521
gamma3_hat 100 0 1.011 0.142 0.715 1.001 1.375
theta_hat 100 0 0.098 0.220 −0.341 0.084 0.571
theta_hat_t_stats 100 0 0.381 0.963 −1.942 0.392 2.439
R2_demand 100 0 0.563 0.146 0.035 0.586 0.793
R2_supply 100 0 0.957 0.011 0.915 0.958 0.977
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.963 0.437 9.013 9.902 11.204
alpha1_hat 100 0 −3.256 26.239 −73.104 −0.447 61.611
alpha2_hat 100 0 20.285 3.231 13.759 20.306 30.019
alpha3_hat 100 0 0.998 0.111 0.765 0.996 1.329
gamma0_hat 100 0 0.929 0.673 −0.988 0.894 2.180
gamma1_hat 100 0 0.661 24.494 −77.505 −0.298 98.402
gamma2_hat 100 0 1.009 0.098 0.736 1.014 1.221
gamma3_hat 100 0 1.016 0.103 0.789 1.010 1.261
theta_hat 100 0 0.113 0.146 −0.378 0.121 0.502
theta_hat_t_stats 100 0 0.671 0.910 −2.384 0.731 2.716
R2_demand 100 0 0.594 0.078 0.364 0.594 0.772
R2_supply 100 0 0.957 0.007 0.934 0.958 0.972
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.995 0.207 9.460 10.015 10.476
alpha1_hat 100 0 0.251 10.457 −23.319 2.316 21.523
alpha2_hat 100 0 20.076 1.288 16.737 19.948 23.231
alpha3_hat 100 0 0.999 0.049 0.851 0.997 1.119
gamma0_hat 100 0 0.926 0.347 0.042 0.883 1.556
gamma1_hat 100 0 1.692 10.652 −19.769 1.393 33.092
gamma2_hat 100 0 1.011 0.049 0.918 1.004 1.140
gamma3_hat 100 0 1.013 0.043 0.903 1.013 1.136
theta_hat 100 0 0.104 0.063 −0.039 0.113 0.223
theta_hat_t_stats 100 0 1.505 0.888 −0.547 1.621 3.152
R2_demand 100 0 0.595 0.039 0.461 0.595 0.667
R2_supply 100 0 0.957 0.003 0.948 0.957 0.965
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.992 0.142 9.640 9.988 10.326
alpha1_hat 100 0 0.453 8.219 −19.675 0.204 15.929
alpha2_hat 100 0 20.017 0.956 17.485 20.057 22.606
alpha3_hat 100 0 0.995 0.034 0.903 0.992 1.085
gamma0_hat 100 0 1.018 0.211 0.483 1.011 1.452
gamma1_hat 100 0 1.081 8.878 −18.857 −0.398 24.715
gamma2_hat 100 0 0.993 0.029 0.923 0.995 1.076
gamma3_hat 100 0 1.002 0.031 0.938 1.002 1.084
theta_hat 100 0 0.101 0.051 −0.045 0.098 0.224
theta_hat_t_stats 100 0 2.088 0.999 −0.978 2.101 4.154
R2_demand 100 0 0.597 0.026 0.518 0.600 0.642
R2_supply 100 0 0.957 0.002 0.952 0.957 0.962
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.003 0.091 9.785 9.992 10.190
alpha1_hat 100 0 0.643 4.690 −10.223 0.769 11.030
alpha2_hat 100 0 20.054 0.586 18.926 20.051 21.517
alpha3_hat 100 0 1.001 0.020 0.945 1.003 1.045
gamma0_hat 100 0 1.022 0.147 0.684 1.017 1.368
gamma1_hat 100 0 2.030 4.785 −9.281 1.683 19.605
gamma2_hat 100 0 0.996 0.022 0.954 0.997 1.055
gamma3_hat 100 0 0.999 0.021 0.952 0.998 1.058
theta_hat 100 0 0.093 0.030 0.004 0.097 0.158
theta_hat_t_stats 100 0 3.106 0.956 0.132 3.201 5.214
R2_demand 100 0 0.594 0.017 0.556 0.596 0.628
R2_supply 100 0 0.957 0.002 0.953 0.957 0.960

\(\theta = 0.2\)

\(\alpha_2 = 0.1\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.075 1.011 7.398 10.061 13.437
alpha1_hat 100 0 1.091 0.872 −0.569 1.004 5.993
alpha2_hat 100 0 0.095 0.088 −0.270 0.097 0.283
alpha3_hat 100 0 1.004 0.238 0.588 0.981 1.694
gamma0_hat 100 0 0.548 1.504 −4.717 0.941 3.049
gamma1_hat 100 0 2.574 22.776 −74.725 0.963 166.484
gamma2_hat 100 0 1.045 0.217 0.601 1.008 1.542
gamma3_hat 100 0 1.037 0.220 0.567 0.996 1.582
theta_hat 100 0 −0.708 13.455 −106.414 0.266 40.890
theta_hat_t_stats 100 0 0.272 1.032 −2.096 0.262 2.472
R2_demand 100 0 0.229 0.389 −1.627 0.339 0.609
R2_supply 100 0 0.976 0.009 0.945 0.977 0.991
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.926 0.661 8.653 9.862 11.727
alpha1_hat 100 0 0.794 0.635 −0.550 0.785 2.855
alpha2_hat 100 0 0.117 0.064 −0.036 0.113 0.357
alpha3_hat 100 0 1.000 0.145 0.676 0.988 1.460
gamma0_hat 100 0 0.863 0.947 −1.332 1.010 2.758
gamma1_hat 100 0 0.879 2.071 −9.676 0.990 11.014
gamma2_hat 100 0 1.017 0.143 0.765 1.001 1.382
gamma3_hat 100 0 1.030 0.141 0.629 1.032 1.427
theta_hat 100 0 0.277 1.101 −5.315 0.223 5.698
theta_hat_t_stats 100 0 0.385 0.951 −1.791 0.365 2.645
R2_demand 100 0 0.269 0.185 −0.489 0.302 0.527
R2_supply 100 0 0.977 0.006 0.960 0.978 0.987
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.959 0.434 8.882 9.918 11.708
alpha1_hat 100 0 0.941 0.458 −0.074 0.949 2.347
alpha2_hat 100 0 0.104 0.043 −0.022 0.102 0.237
alpha3_hat 100 0 1.001 0.102 0.755 0.998 1.358
gamma0_hat 100 0 0.948 0.712 −1.078 0.975 2.415
gamma1_hat 100 0 0.951 1.194 −2.037 1.051 7.303
gamma2_hat 100 0 1.005 0.112 0.712 1.001 1.284
gamma3_hat 100 0 1.007 0.102 0.747 1.012 1.239
theta_hat 100 0 0.256 0.636 −2.813 0.175 2.333
theta_hat_t_stats 100 0 0.486 1.017 −2.527 0.456 2.639
R2_demand 100 0 0.264 0.137 −0.149 0.274 0.532
R2_supply 100 0 0.977 0.004 0.963 0.977 0.985
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.031 0.176 9.642 10.023 10.461
alpha1_hat 100 0 1.010 0.190 0.643 1.021 1.701
alpha2_hat 100 0 0.101 0.019 0.043 0.098 0.152
alpha3_hat 100 0 1.008 0.037 0.932 1.008 1.111
gamma0_hat 100 0 1.029 0.337 0.179 1.029 2.079
gamma1_hat 100 0 1.003 0.370 −0.138 1.008 1.761
gamma2_hat 100 0 0.993 0.051 0.846 0.990 1.116
gamma3_hat 100 0 0.995 0.046 0.880 0.995 1.112
theta_hat 100 0 0.196 0.191 −0.243 0.175 0.747
theta_hat_t_stats 100 0 0.998 0.913 −1.285 1.012 3.137
R2_demand 100 0 0.264 0.056 0.134 0.271 0.373
R2_supply 100 0 0.977 0.002 0.973 0.977 0.981
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.008 0.132 9.671 10.024 10.394
alpha1_hat 100 0 1.011 0.132 0.646 1.015 1.329
alpha2_hat 100 0 0.099 0.012 0.069 0.098 0.137
alpha3_hat 100 0 0.999 0.030 0.918 1.001 1.077
gamma0_hat 100 0 0.952 0.212 0.394 0.966 1.475
gamma1_hat 100 0 0.962 0.256 0.461 0.970 1.765
gamma2_hat 100 0 1.005 0.030 0.929 1.004 1.069
gamma3_hat 100 0 1.005 0.031 0.925 1.003 1.075
theta_hat 100 0 0.229 0.137 −0.166 0.216 0.521
theta_hat_t_stats 100 0 1.695 0.961 −1.275 1.658 3.443
R2_demand 100 0 0.276 0.042 0.161 0.276 0.375
R2_supply 100 0 0.977 0.001 0.973 0.977 0.979
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.995 0.092 9.780 10.005 10.230
alpha1_hat 100 0 0.988 0.087 0.790 0.990 1.212
alpha2_hat 100 0 0.101 0.008 0.081 0.101 0.120
alpha3_hat 100 0 0.999 0.021 0.942 0.998 1.052
gamma0_hat 100 0 0.989 0.114 0.707 0.989 1.246
gamma1_hat 100 0 0.998 0.155 0.653 0.994 1.375
gamma2_hat 100 0 1.003 0.018 0.962 1.002 1.048
gamma3_hat 100 0 1.001 0.017 0.958 1.000 1.043
theta_hat 100 0 0.203 0.074 0.025 0.200 0.394
theta_hat_t_stats 100 0 2.483 0.886 0.295 2.456 4.902
R2_demand 100 0 0.279 0.026 0.191 0.279 0.337
R2_supply 100 0 0.977 0.001 0.975 0.977 0.979

\(\alpha_2 = 0.5\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.931 0.852 7.610 9.913 12.363
alpha1_hat 100 0 0.945 2.071 −5.565 0.842 6.752
alpha2_hat 100 0 0.494 0.218 −0.055 0.491 1.175
alpha3_hat 100 0 0.991 0.212 0.508 0.965 1.470
gamma0_hat 100 0 0.902 1.615 −6.741 1.067 4.386
gamma1_hat 100 0 0.035 4.734 −35.358 0.795 6.217
gamma2_hat 100 0 0.995 0.212 0.588 0.980 1.714
gamma3_hat 100 0 0.989 0.210 0.510 0.993 1.529
theta_hat 100 0 0.415 0.873 −0.539 0.234 5.647
theta_hat_t_stats 100 0 0.499 0.890 −1.713 0.593 2.092
R2_demand 100 0 0.366 0.272 −0.637 0.422 0.733
R2_supply 100 0 0.973 0.012 0.899 0.975 0.992
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.000 0.609 8.668 9.966 11.644
alpha1_hat 100 0 0.873 1.411 −2.676 0.841 4.144
alpha2_hat 100 0 0.514 0.145 0.207 0.512 1.042
alpha3_hat 100 0 1.013 0.139 0.662 1.010 1.360
gamma0_hat 100 0 0.886 1.115 −2.705 1.022 2.746
gamma1_hat 100 0 0.820 1.754 −4.287 0.737 5.277
gamma2_hat 100 0 1.015 0.163 0.761 0.999 1.562
gamma3_hat 100 0 1.008 0.169 0.710 0.971 1.518
theta_hat 100 0 0.249 0.361 −0.511 0.216 1.569
theta_hat_t_stats 100 0 0.703 0.940 −1.470 0.733 3.272
R2_demand 100 0 0.411 0.176 −0.371 0.446 0.713
R2_supply 100 0 0.970 0.008 0.934 0.971 0.985
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.971 0.392 9.083 9.932 11.142
alpha1_hat 100 0 0.983 1.008 −1.916 1.021 3.446
alpha2_hat 100 0 0.498 0.106 0.255 0.489 0.800
alpha3_hat 100 0 0.983 0.095 0.814 0.981 1.286
gamma0_hat 100 0 0.755 0.696 −1.330 0.714 2.564
gamma1_hat 100 0 0.811 1.349 −2.427 0.846 4.138
gamma2_hat 100 0 1.028 0.098 0.814 1.018 1.310
gamma3_hat 100 0 1.031 0.095 0.803 1.030 1.332
theta_hat 100 0 0.273 0.238 −0.289 0.268 0.956
theta_hat_t_stats 100 0 1.127 0.927 −1.723 1.083 3.000
R2_demand 100 0 0.426 0.108 0.137 0.451 0.643
R2_supply 100 0 0.969 0.006 0.947 0.970 0.981
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.028 0.204 9.613 10.024 10.608
alpha1_hat 100 0 1.011 0.439 0.159 1.018 2.029
alpha2_hat 100 0 0.503 0.053 0.384 0.510 0.622
alpha3_hat 100 0 1.010 0.046 0.915 1.012 1.112
gamma0_hat 100 0 0.976 0.312 0.207 0.961 2.034
gamma1_hat 100 0 0.925 0.569 −0.508 0.882 2.970
gamma2_hat 100 0 1.002 0.047 0.860 1.003 1.134
gamma3_hat 100 0 1.003 0.042 0.882 1.001 1.118
theta_hat 100 0 0.215 0.110 −0.181 0.202 0.513
theta_hat_t_stats 100 0 2.230 1.047 −1.911 2.222 5.614
R2_demand 100 0 0.425 0.045 0.296 0.425 0.511
R2_supply 100 0 0.970 0.003 0.961 0.970 0.977
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.001 0.135 9.705 9.999 10.503
alpha1_hat 100 0 1.015 0.304 0.296 1.053 1.931
alpha2_hat 100 0 0.499 0.031 0.434 0.495 0.592
alpha3_hat 100 0 1.002 0.030 0.923 1.002 1.124
gamma0_hat 100 0 0.988 0.211 0.491 1.000 1.475
gamma1_hat 100 0 0.940 0.370 0.050 0.914 1.805
gamma2_hat 100 0 1.000 0.030 0.932 0.998 1.068
gamma3_hat 100 0 1.001 0.030 0.912 1.002 1.077
theta_hat 100 0 0.213 0.065 0.038 0.224 0.324
theta_hat_t_stats 100 0 3.165 0.942 0.601 3.287 4.820
R2_demand 100 0 0.429 0.035 0.313 0.426 0.516
R2_supply 100 0 0.970 0.002 0.965 0.970 0.974
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.001 0.086 9.814 9.999 10.170
alpha1_hat 100 0 1.027 0.208 0.573 1.011 1.535
alpha2_hat 100 0 0.498 0.024 0.444 0.498 0.558
alpha3_hat 100 0 1.001 0.021 0.959 0.999 1.056
gamma0_hat 100 0 1.006 0.146 0.594 1.010 1.400
gamma1_hat 100 0 0.974 0.254 0.304 0.962 1.555
gamma2_hat 100 0 0.999 0.021 0.954 0.999 1.059
gamma3_hat 100 0 0.997 0.022 0.943 0.998 1.061
theta_hat 100 0 0.204 0.041 0.113 0.207 0.299
theta_hat_t_stats 100 0 4.843 0.938 2.695 4.903 7.199
R2_demand 100 0 0.427 0.022 0.380 0.429 0.481
R2_supply 100 0 0.970 0.001 0.967 0.970 0.973

\(\alpha_2 = 1.0\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.040 1.033 8.325 9.983 14.154
alpha1_hat 100 0 0.732 3.527 −7.984 0.989 13.497
alpha2_hat 100 0 1.034 0.395 −0.040 0.982 2.165
alpha3_hat 100 0 1.018 0.228 0.547 1.015 2.066
gamma0_hat 100 0 0.685 1.396 −3.283 0.908 3.717
gamma1_hat 100 0 −0.548 14.792 −137.315 1.171 18.350
gamma2_hat 100 0 1.035 0.209 0.596 1.011 1.701
gamma3_hat 100 0 1.052 0.237 0.534 1.020 1.689
theta_hat 100 0 0.374 1.192 −1.655 0.165 10.772
theta_hat_t_stats 100 0 0.539 1.091 −2.646 0.446 3.378
R2_demand 100 0 0.442 0.271 −1.045 0.510 0.773
R2_supply 100 0 0.967 0.013 0.935 0.969 0.988
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.888 0.665 8.647 9.810 11.863
alpha1_hat 100 0 1.235 2.322 −4.598 1.410 5.977
alpha2_hat 100 0 0.950 0.277 0.235 0.969 1.804
alpha3_hat 100 0 0.979 0.162 0.634 0.967 1.363
gamma0_hat 100 0 0.991 0.985 −1.499 1.034 3.186
gamma1_hat 100 0 0.778 3.913 −17.340 1.335 8.883
gamma2_hat 100 0 0.993 0.143 0.652 0.984 1.396
gamma3_hat 100 0 1.015 0.159 0.660 0.995 1.431
theta_hat 100 0 0.246 0.429 −0.645 0.184 2.547
theta_hat_t_stats 100 0 0.639 1.071 −1.637 0.676 3.217
R2_demand 100 0 0.474 0.143 −0.081 0.508 0.740
R2_supply 100 0 0.968 0.008 0.943 0.970 0.984
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.989 0.402 9.067 9.978 10.821
alpha1_hat 100 0 0.892 1.662 −2.511 0.818 5.197
alpha2_hat 100 0 1.010 0.174 0.651 1.021 1.376
alpha3_hat 100 0 1.008 0.093 0.737 1.010 1.240
gamma0_hat 100 0 0.895 0.701 −1.357 0.977 2.249
gamma1_hat 100 0 1.147 1.722 −3.622 1.101 5.098
gamma2_hat 100 0 1.018 0.107 0.777 1.008 1.352
gamma3_hat 100 0 1.013 0.098 0.841 1.005 1.347
theta_hat 100 0 0.198 0.183 −0.195 0.210 0.722
theta_hat_t_stats 100 0 0.970 0.895 −0.912 1.048 2.782
R2_demand 100 0 0.475 0.105 0.002 0.487 0.678
R2_supply 100 0 0.967 0.006 0.937 0.968 0.980
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.988 0.171 9.578 9.975 10.431
alpha1_hat 100 0 0.821 0.715 −0.816 0.789 2.254
alpha2_hat 100 0 1.015 0.078 0.850 1.014 1.272
alpha3_hat 100 0 0.997 0.042 0.893 1.000 1.083
gamma0_hat 100 0 0.963 0.309 0.199 0.954 1.754
gamma1_hat 100 0 1.064 0.759 −0.706 1.039 3.020
gamma2_hat 100 0 1.004 0.045 0.875 1.004 1.105
gamma3_hat 100 0 1.003 0.044 0.893 1.005 1.115
theta_hat 100 0 0.201 0.078 0.025 0.204 0.366
theta_hat_t_stats 100 0 2.413 0.878 0.311 2.459 4.222
R2_demand 100 0 0.473 0.043 0.342 0.479 0.554
R2_supply 100 0 0.967 0.003 0.961 0.967 0.973
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.996 0.145 9.645 9.980 10.376
alpha1_hat 100 0 1.038 0.491 0.048 1.017 2.282
alpha2_hat 100 0 0.995 0.058 0.795 0.986 1.122
alpha3_hat 100 0 1.000 0.032 0.908 1.001 1.091
gamma0_hat 100 0 0.962 0.217 0.315 0.979 1.463
gamma1_hat 100 0 1.026 0.641 −0.504 1.069 2.565
gamma2_hat 100 0 1.004 0.034 0.933 1.000 1.087
gamma3_hat 100 0 1.001 0.031 0.910 1.003 1.110
theta_hat 100 0 0.204 0.064 0.042 0.199 0.377
theta_hat_t_stats 100 0 3.397 0.972 0.804 3.354 5.773
R2_demand 100 0 0.479 0.035 0.380 0.483 0.534
R2_supply 100 0 0.967 0.002 0.963 0.967 0.971
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.992 0.080 9.850 9.989 10.241
alpha1_hat 100 0 0.987 0.308 0.308 0.955 2.050
alpha2_hat 100 0 0.999 0.032 0.921 1.004 1.083
alpha3_hat 100 0 0.999 0.020 0.950 0.998 1.046
gamma0_hat 100 0 1.013 0.133 0.654 1.009 1.301
gamma1_hat 100 0 1.041 0.344 0.096 1.099 1.727
gamma2_hat 100 0 1.000 0.020 0.963 0.998 1.048
gamma3_hat 100 0 0.997 0.020 0.939 0.998 1.049
theta_hat 100 0 0.194 0.034 0.114 0.196 0.285
theta_hat_t_stats 100 0 5.231 0.796 3.209 5.268 7.079
R2_demand 100 0 0.472 0.020 0.413 0.475 0.513
R2_supply 100 0 0.968 0.001 0.964 0.967 0.970

\(\alpha_2 = 5.0\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.002 1.058 7.983 9.898 14.800
alpha1_hat 100 0 1.377 15.017 −48.230 0.894 38.341
alpha2_hat 100 0 4.940 1.918 0.544 4.849 15.191
alpha3_hat 100 0 1.006 0.242 0.572 0.991 2.311
gamma0_hat 100 0 0.511 1.736 −5.222 0.866 3.351
gamma1_hat 100 0 −4.801 27.217 −157.492 1.365 32.063
gamma2_hat 100 0 1.067 0.272 0.558 1.023 2.006
gamma3_hat 100 0 1.058 0.206 0.657 1.021 1.673
theta_hat 100 0 0.391 0.651 −0.854 0.264 3.853
theta_hat_t_stats 100 0 0.655 0.933 −3.120 0.694 2.744
R2_demand 100 0 0.506 0.292 −1.126 0.570 0.816
R2_supply 100 0 0.962 0.016 0.886 0.965 0.985
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.019 0.671 8.679 10.062 12.231
alpha1_hat 100 0 2.664 10.353 −18.538 1.161 33.037
alpha2_hat 100 0 4.837 1.178 1.936 4.857 7.383
alpha3_hat 100 0 1.002 0.139 0.698 0.996 1.357
gamma0_hat 100 0 0.999 0.967 −1.678 1.054 3.016
gamma1_hat 100 0 −0.080 11.572 −49.144 0.659 23.910
gamma2_hat 100 0 0.993 0.160 0.569 0.981 1.481
gamma3_hat 100 0 0.997 0.155 0.681 0.995 1.343
theta_hat 100 0 0.235 0.280 −0.298 0.192 1.542
theta_hat_t_stats 100 0 0.775 0.816 −1.355 0.739 2.769
R2_demand 100 0 0.507 0.151 −0.049 0.532 0.741
R2_supply 100 0 0.965 0.009 0.935 0.967 0.982
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.955 0.387 9.235 9.913 11.116
alpha1_hat 100 0 0.953 6.612 −20.606 1.011 18.557
alpha2_hat 100 0 4.949 0.771 3.250 4.930 7.343
alpha3_hat 100 0 0.980 0.087 0.739 0.973 1.348
gamma0_hat 100 0 0.897 0.786 −1.441 1.018 2.274
gamma1_hat 100 0 0.073 7.274 −23.077 0.463 18.307
gamma2_hat 100 0 1.013 0.114 0.813 0.995 1.324
gamma3_hat 100 0 1.007 0.110 0.743 1.009 1.265
theta_hat 100 0 0.238 0.176 −0.171 0.218 0.814
theta_hat_t_stats 100 0 1.266 0.860 −1.171 1.262 2.948
R2_demand 100 0 0.521 0.102 0.148 0.532 0.682
R2_supply 100 0 0.964 0.008 0.941 0.964 0.975
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.982 0.193 9.560 9.971 10.435
alpha1_hat 100 0 1.092 3.330 −6.629 0.974 7.706
alpha2_hat 100 0 4.972 0.378 4.148 5.007 5.920
alpha3_hat 100 0 0.992 0.049 0.891 0.986 1.130
gamma0_hat 100 0 0.945 0.330 −0.030 0.942 1.641
gamma1_hat 100 0 0.908 3.676 −7.279 0.801 10.889
gamma2_hat 100 0 1.006 0.047 0.896 1.008 1.126
gamma3_hat 100 0 1.013 0.047 0.900 1.010 1.109
theta_hat 100 0 0.211 0.078 0.001 0.211 0.369
theta_hat_t_stats 100 0 2.746 0.945 0.017 2.874 4.744
R2_demand 100 0 0.521 0.039 0.432 0.517 0.609
R2_supply 100 0 0.964 0.003 0.956 0.964 0.970
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.999 0.134 9.692 10.002 10.278
alpha1_hat 100 0 1.050 2.303 −6.558 1.026 6.791
alpha2_hat 100 0 4.995 0.248 4.188 4.964 5.820
alpha3_hat 100 0 1.002 0.030 0.929 1.001 1.072
gamma0_hat 100 0 0.983 0.216 0.423 0.979 1.454
gamma1_hat 100 0 0.878 2.265 −4.816 0.770 6.018
gamma2_hat 100 0 1.003 0.033 0.915 1.000 1.104
gamma3_hat 100 0 0.995 0.031 0.916 1.001 1.074
theta_hat 100 0 0.205 0.052 0.093 0.209 0.337
theta_hat_t_stats 100 0 3.838 0.892 1.855 3.918 6.058
R2_demand 100 0 0.519 0.029 0.437 0.521 0.587
R2_supply 100 0 0.964 0.002 0.958 0.964 0.970
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.000 0.084 9.818 10.003 10.194
alpha1_hat 100 0 0.896 1.299 −2.603 1.018 4.193
alpha2_hat 100 0 5.009 0.169 4.558 5.006 5.406
alpha3_hat 100 0 1.001 0.019 0.951 1.001 1.055
gamma0_hat 100 0 1.001 0.123 0.673 1.007 1.323
gamma1_hat 100 0 1.069 1.455 −2.417 1.146 4.260
gamma2_hat 100 0 0.999 0.020 0.956 0.999 1.052
gamma3_hat 100 0 0.998 0.018 0.957 0.999 1.037
theta_hat 100 0 0.199 0.035 0.110 0.196 0.314
theta_hat_t_stats 100 0 5.950 0.948 3.341 5.883 8.748
R2_demand 100 0 0.519 0.017 0.475 0.520 0.551
R2_supply 100 0 0.964 0.001 0.962 0.964 0.967

\(\alpha_2 = 20.0\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.907 0.938 7.866 9.809 12.850
alpha1_hat 100 0 1.370 51.562 −120.993 −5.489 170.972
alpha2_hat 100 0 19.581 6.126 3.832 19.461 40.350
alpha3_hat 100 0 0.994 0.232 0.524 0.955 1.846
gamma0_hat 100 0 0.408 2.155 −9.663 0.710 4.317
gamma1_hat 100 0 6.310 89.940 −415.311 10.375 192.299
gamma2_hat 100 0 1.058 0.282 0.548 1.047 2.227
gamma3_hat 100 0 1.080 0.347 0.632 1.011 2.963
theta_hat 100 0 0.262 0.494 −0.626 0.172 2.440
theta_hat_t_stats 100 0 0.423 0.913 −2.026 0.516 2.709
R2_demand 100 0 0.504 0.230 −0.385 0.553 0.813
R2_supply 100 0 0.958 0.030 0.748 0.965 0.985
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.964 0.590 8.849 9.864 12.076
alpha1_hat 100 0 −6.842 38.993 −106.509 −8.018 94.391
alpha2_hat 100 0 20.716 4.666 11.087 20.285 34.440
alpha3_hat 100 0 0.994 0.129 0.703 0.993 1.496
gamma0_hat 100 0 1.017 1.082 −2.746 1.186 2.885
gamma1_hat 100 0 −0.088 44.747 −150.445 1.305 160.906
gamma2_hat 100 0 1.003 0.161 0.668 0.977 1.508
gamma3_hat 100 0 0.978 0.148 0.646 0.972 1.490
theta_hat 100 0 0.206 0.265 −0.593 0.154 1.063
theta_hat_t_stats 100 0 0.767 0.882 −1.602 0.700 2.806
R2_demand 100 0 0.526 0.168 −0.418 0.558 0.786
R2_supply 100 0 0.965 0.009 0.932 0.966 0.979
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.041 0.450 9.145 10.047 11.300
alpha1_hat 100 0 3.898 32.444 −76.175 2.590 80.157
alpha2_hat 100 0 19.927 3.680 12.924 19.748 32.442
alpha3_hat 100 0 1.025 0.102 0.792 1.028 1.270
gamma0_hat 100 0 1.040 0.623 −0.460 1.052 2.627
gamma1_hat 100 0 −4.063 25.418 −65.590 −4.101 47.477
gamma2_hat 100 0 0.990 0.095 0.707 0.983 1.242
gamma3_hat 100 0 1.003 0.099 0.771 0.997 1.234
theta_hat 100 0 0.222 0.151 −0.115 0.210 0.605
theta_hat_t_stats 100 0 1.242 0.758 −0.918 1.286 2.638
R2_demand 100 0 0.517 0.094 0.251 0.529 0.722
R2_supply 100 0 0.964 0.006 0.941 0.964 0.976
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.986 0.172 9.642 9.959 10.463
alpha1_hat 100 0 1.846 13.554 −32.313 1.997 35.088
alpha2_hat 100 0 19.849 1.481 16.789 19.801 23.422
alpha3_hat 100 0 0.995 0.044 0.904 0.992 1.124
gamma0_hat 100 0 1.037 0.276 0.146 1.034 1.619
gamma1_hat 100 0 −0.142 13.378 −30.588 −1.968 38.986
gamma2_hat 100 0 0.997 0.040 0.914 0.993 1.104
gamma3_hat 100 0 0.998 0.046 0.907 0.993 1.134
theta_hat 100 0 0.202 0.076 0.026 0.211 0.375
theta_hat_t_stats 100 0 2.699 0.965 0.373 2.843 5.085
R2_demand 100 0 0.531 0.041 0.385 0.536 0.609
R2_supply 100 0 0.964 0.002 0.954 0.964 0.970
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.989 0.154 9.663 9.990 10.370
alpha1_hat 100 0 2.484 8.549 −26.399 3.687 21.425
alpha2_hat 100 0 19.802 1.119 17.548 19.748 23.647
alpha3_hat 100 0 0.999 0.034 0.918 1.000 1.075
gamma0_hat 100 0 0.989 0.192 0.541 0.985 1.465
gamma1_hat 100 0 1.404 9.574 −24.190 1.093 20.914
gamma2_hat 100 0 1.003 0.029 0.919 1.002 1.087
gamma3_hat 100 0 0.998 0.031 0.928 1.000 1.061
theta_hat 100 0 0.199 0.053 0.077 0.196 0.329
theta_hat_t_stats 100 0 3.772 0.896 1.580 3.739 5.662
R2_demand 100 0 0.530 0.029 0.470 0.531 0.594
R2_supply 100 0 0.963 0.002 0.958 0.963 0.968
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.007 0.092 9.755 10.000 10.283
alpha1_hat 100 0 0.720 5.440 −13.112 0.422 12.155
alpha2_hat 100 0 20.050 0.645 18.084 20.056 21.485
alpha3_hat 100 0 1.003 0.022 0.949 1.003 1.072
gamma0_hat 100 0 0.989 0.132 0.639 0.993 1.273
gamma1_hat 100 0 1.541 5.463 −10.233 1.874 11.974
gamma2_hat 100 0 1.000 0.019 0.958 1.000 1.054
gamma3_hat 100 0 1.001 0.020 0.961 1.001 1.060
theta_hat 100 0 0.199 0.033 0.111 0.198 0.281
theta_hat_t_stats 100 0 6.074 0.914 3.798 6.052 8.207
R2_demand 100 0 0.530 0.021 0.467 0.532 0.578
R2_supply 100 0 0.963 0.001 0.960 0.963 0.967

\(\theta = 0.33\)

\(\alpha_2 = 0.1\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.101 0.903 7.916 10.022 12.702
alpha1_hat 100 0 1.053 1.063 −1.100 0.896 4.684
alpha2_hat 100 0 0.102 0.093 −0.171 0.098 0.323
alpha3_hat 100 0 1.003 0.219 0.569 0.991 1.623
gamma0_hat 100 0 1.065 1.516 −4.142 1.438 4.185
gamma1_hat 100 0 4.401 33.922 −15.564 1.224 337.594
gamma2_hat 100 0 0.982 0.211 0.382 0.949 1.643
gamma3_hat 100 0 0.981 0.200 0.535 0.958 1.596
theta_hat 100 0 −1.026 13.210 −129.878 0.127 9.248
theta_hat_t_stats 100 0 0.150 0.935 −2.170 0.180 2.524
R2_demand 100 0 0.169 0.305 −0.935 0.247 0.609
R2_supply 100 0 0.979 0.009 0.947 0.981 0.992
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.942 0.645 8.511 9.888 11.803
alpha1_hat 100 0 1.035 0.659 −1.098 1.009 2.949
alpha2_hat 100 0 0.094 0.060 −0.039 0.092 0.305
alpha3_hat 100 0 1.003 0.158 0.666 0.989 1.558
gamma0_hat 100 0 1.035 1.136 −2.674 1.152 2.769
gamma1_hat 100 0 1.637 6.820 −14.264 0.813 52.629
gamma2_hat 100 0 0.995 0.166 0.703 0.966 1.457
gamma3_hat 100 0 0.999 0.160 0.665 0.985 1.541
theta_hat 100 0 −0.004 3.435 −26.394 0.403 8.146
theta_hat_t_stats 100 0 0.438 0.969 −1.833 0.528 2.155
R2_demand 100 0 0.245 0.153 −0.357 0.283 0.542
R2_supply 100 0 0.979 0.006 0.959 0.980 0.988
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.021 0.418 9.172 9.970 11.266
alpha1_hat 100 0 1.039 0.462 −0.247 0.990 2.050
alpha2_hat 100 0 0.097 0.042 −0.003 0.096 0.235
alpha3_hat 100 0 1.008 0.096 0.801 1.004 1.253
gamma0_hat 100 0 0.854 0.652 −0.964 0.852 2.344
gamma1_hat 100 0 0.661 1.543 −11.142 0.899 4.071
gamma2_hat 100 0 1.014 0.094 0.837 1.006 1.252
gamma3_hat 100 0 1.026 0.101 0.817 1.024 1.278
theta_hat 100 0 0.529 0.745 −0.976 0.379 6.205
theta_hat_t_stats 100 0 0.911 0.748 −0.853 0.832 2.497
R2_demand 100 0 0.230 0.134 −0.151 0.238 0.537
R2_supply 100 0 0.978 0.004 0.965 0.979 0.985
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.987 0.203 9.377 9.972 10.663
alpha1_hat 100 0 1.005 0.200 0.332 1.003 1.444
alpha2_hat 100 0 0.099 0.020 0.053 0.101 0.155
alpha3_hat 100 0 1.004 0.048 0.843 1.004 1.118
gamma0_hat 100 0 1.011 0.300 0.171 1.027 1.736
gamma1_hat 100 0 0.959 0.411 −0.356 0.954 1.920
gamma2_hat 100 0 0.995 0.050 0.861 0.996 1.121
gamma3_hat 100 0 1.008 0.038 0.926 1.003 1.103
theta_hat 100 0 0.355 0.213 −0.201 0.351 1.184
theta_hat_t_stats 100 0 1.713 0.967 −1.073 1.808 4.122
R2_demand 100 0 0.241 0.058 0.101 0.249 0.371
R2_supply 100 0 0.978 0.002 0.975 0.978 0.982
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.010 0.136 9.632 10.018 10.284
alpha1_hat 100 0 1.010 0.130 0.719 1.001 1.343
alpha2_hat 100 0 0.099 0.014 0.067 0.098 0.142
alpha3_hat 100 0 0.998 0.033 0.931 0.999 1.087
gamma0_hat 100 0 1.007 0.192 0.438 1.037 1.375
gamma1_hat 100 0 0.948 0.311 0.071 0.921 1.874
gamma2_hat 100 0 0.998 0.031 0.924 0.998 1.103
gamma3_hat 100 0 0.998 0.027 0.943 0.998 1.076
theta_hat 100 0 0.354 0.156 −0.140 0.352 0.801
theta_hat_t_stats 100 0 2.481 1.054 −1.044 2.525 4.915
R2_demand 100 0 0.230 0.042 0.126 0.223 0.333
R2_supply 100 0 0.979 0.001 0.975 0.979 0.981
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.996 0.094 9.756 10.009 10.186
alpha1_hat 100 0 0.984 0.093 0.700 0.981 1.208
alpha2_hat 100 0 0.101 0.009 0.079 0.101 0.121
alpha3_hat 100 0 1.001 0.021 0.948 1.002 1.060
gamma0_hat 100 0 0.993 0.133 0.602 0.990 1.271
gamma1_hat 100 0 1.052 0.177 0.580 1.046 1.477
gamma2_hat 100 0 1.001 0.020 0.951 1.002 1.051
gamma3_hat 100 0 1.000 0.019 0.959 0.999 1.044
theta_hat 100 0 0.307 0.090 0.048 0.298 0.540
theta_hat_t_stats 100 0 3.495 0.974 0.639 3.565 5.808
R2_demand 100 0 0.240 0.027 0.161 0.242 0.292
R2_supply 100 0 0.978 0.001 0.976 0.978 0.980

\(\alpha_2 = 0.5\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.788 0.869 7.639 9.751 13.200
alpha1_hat 100 0 0.846 2.077 −3.651 0.938 6.905
alpha2_hat 100 0 0.484 0.233 −0.058 0.490 1.293
alpha3_hat 100 0 0.971 0.219 0.492 0.951 1.737
gamma0_hat 100 0 1.061 1.365 −2.315 1.101 4.408
gamma1_hat 100 0 0.042 22.643 −146.792 1.182 118.213
gamma2_hat 100 0 1.002 0.196 0.603 0.988 1.425
gamma3_hat 100 0 0.992 0.190 0.624 0.982 1.567
theta_hat 100 0 0.513 4.245 −17.446 0.260 27.957
theta_hat_t_stats 100 0 0.510 0.974 −3.851 0.583 2.632
R2_demand 100 0 0.366 0.273 −1.213 0.436 0.762
R2_supply 100 0 0.975 0.009 0.943 0.977 0.990
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.124 0.601 8.625 10.097 11.521
alpha1_hat 100 0 0.972 1.435 −3.407 0.881 5.203
alpha2_hat 100 0 0.523 0.162 0.116 0.529 0.973
alpha3_hat 100 0 1.044 0.148 0.716 1.051 1.519
gamma0_hat 100 0 1.200 0.851 −0.579 1.262 3.107
gamma1_hat 100 0 1.052 1.975 −4.719 1.070 7.520
gamma2_hat 100 0 0.985 0.126 0.697 0.989 1.263
gamma3_hat 100 0 0.976 0.136 0.557 0.973 1.354
theta_hat 100 0 0.282 0.345 −0.644 0.258 1.505
theta_hat_t_stats 100 0 0.783 0.806 −1.187 0.956 2.203
R2_demand 100 0 0.314 0.194 −0.334 0.351 0.618
R2_supply 100 0 0.976 0.006 0.957 0.977 0.988
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.969 0.451 8.923 9.933 11.168
alpha1_hat 100 0 0.994 1.062 −2.433 0.846 4.063
alpha2_hat 100 0 0.496 0.109 0.221 0.488 0.786
alpha3_hat 100 0 0.999 0.116 0.726 1.009 1.273
gamma0_hat 100 0 1.154 0.607 −0.438 1.157 2.422
gamma1_hat 100 0 0.901 1.635 −3.139 0.952 4.285
gamma2_hat 100 0 0.979 0.087 0.767 0.975 1.239
gamma3_hat 100 0 0.997 0.105 0.723 0.989 1.348
theta_hat 100 0 0.329 0.285 −0.214 0.290 1.076
theta_hat_t_stats 100 0 1.306 1.048 −0.894 1.256 3.955
R2_demand 100 0 0.357 0.115 0.000 0.379 0.556
R2_supply 100 0 0.974 0.005 0.960 0.975 0.982
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.989 0.216 9.527 9.985 10.570
alpha1_hat 100 0 0.921 0.428 −0.146 0.935 2.314
alpha2_hat 100 0 0.506 0.054 0.332 0.504 0.620
alpha3_hat 100 0 1.000 0.042 0.899 1.000 1.119
gamma0_hat 100 0 1.017 0.287 −0.045 1.054 1.589
gamma1_hat 100 0 1.098 0.555 −0.208 1.035 2.450
gamma2_hat 100 0 1.000 0.042 0.899 1.005 1.117
gamma3_hat 100 0 1.000 0.043 0.897 0.994 1.095
theta_hat 100 0 0.311 0.105 0.044 0.306 0.590
theta_hat_t_stats 100 0 3.020 0.907 0.415 3.118 4.864
R2_demand 100 0 0.355 0.058 0.160 0.364 0.475
R2_supply 100 0 0.973 0.002 0.968 0.974 0.978
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.983 0.126 9.699 9.986 10.273
alpha1_hat 100 0 0.951 0.318 0.161 0.927 1.761
alpha2_hat 100 0 0.503 0.034 0.427 0.500 0.587
alpha3_hat 100 0 0.996 0.031 0.916 0.995 1.082
gamma0_hat 100 0 0.978 0.216 0.407 0.992 1.508
gamma1_hat 100 0 1.010 0.402 0.062 1.028 1.965
gamma2_hat 100 0 1.006 0.030 0.932 1.005 1.094
gamma3_hat 100 0 1.000 0.031 0.921 1.002 1.072
theta_hat 100 0 0.332 0.081 0.139 0.331 0.558
theta_hat_t_stats 100 0 4.518 0.972 1.945 4.632 6.625
R2_demand 100 0 0.366 0.031 0.286 0.368 0.439
R2_supply 100 0 0.973 0.001 0.970 0.973 0.977
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.003 0.085 9.817 9.998 10.200
alpha1_hat 100 0 1.034 0.202 0.571 1.027 1.492
alpha2_hat 100 0 0.497 0.022 0.452 0.499 0.546
alpha3_hat 100 0 1.000 0.021 0.946 1.000 1.058
gamma0_hat 100 0 1.011 0.124 0.696 1.016 1.311
gamma1_hat 100 0 0.923 0.271 0.239 0.928 1.799
gamma2_hat 100 0 0.999 0.017 0.948 0.999 1.041
gamma3_hat 100 0 0.997 0.019 0.954 0.998 1.044
theta_hat 100 0 0.341 0.049 0.157 0.344 0.473
theta_hat_t_stats 100 0 7.355 0.955 3.389 7.404 9.223
R2_demand 100 0 0.359 0.021 0.309 0.361 0.424
R2_supply 100 0 0.973 0.001 0.971 0.973 0.975

\(\alpha_2 = 1.0\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.881 0.911 7.719 9.780 12.297
alpha1_hat 100 0 0.820 3.832 −10.976 1.218 10.451
alpha2_hat 100 0 0.987 0.444 0.155 0.988 2.274
alpha3_hat 100 0 0.974 0.233 0.323 0.985 1.659
gamma0_hat 100 0 0.641 1.874 −6.248 0.962 3.918
gamma1_hat 100 0 −0.016 8.935 −31.752 0.069 59.196
gamma2_hat 100 0 1.053 0.269 0.550 1.019 1.986
gamma3_hat 100 0 1.061 0.272 0.609 1.016 2.385
theta_hat 100 0 0.545 0.970 −3.919 0.437 4.710
theta_hat_t_stats 100 0 0.756 0.912 −2.111 0.817 2.723
R2_demand 100 0 0.433 0.229 −0.605 0.487 0.756
R2_supply 100 0 0.970 0.015 0.869 0.974 0.989
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.911 0.619 8.764 9.845 11.944
alpha1_hat 100 0 0.851 2.377 −4.596 0.806 6.231
alpha2_hat 100 0 0.991 0.288 0.353 0.978 1.800
alpha3_hat 100 0 0.999 0.152 0.641 0.992 1.480
gamma0_hat 100 0 0.931 0.933 −2.048 0.944 3.379
gamma1_hat 100 0 1.204 3.438 −6.497 1.582 9.392
gamma2_hat 100 0 1.011 0.142 0.694 0.998 1.433
gamma3_hat 100 0 0.996 0.146 0.687 0.990 1.483
theta_hat 100 0 0.333 0.356 −0.373 0.300 1.199
theta_hat_t_stats 100 0 0.980 0.965 −0.936 0.981 3.298
R2_demand 100 0 0.402 0.166 −0.199 0.436 0.642
R2_supply 100 0 0.972 0.006 0.947 0.973 0.984
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.991 0.399 9.100 9.926 10.992
alpha1_hat 100 0 0.978 1.600 −2.596 1.086 4.918
alpha2_hat 100 0 1.004 0.184 0.606 0.988 1.468
alpha3_hat 100 0 0.998 0.087 0.807 0.986 1.228
gamma0_hat 100 0 1.030 0.696 −0.888 1.017 3.541
gamma1_hat 100 0 1.210 2.091 −6.672 1.261 6.603
gamma2_hat 100 0 0.990 0.104 0.637 0.987 1.256
gamma3_hat 100 0 0.998 0.100 0.708 0.998 1.296
theta_hat 100 0 0.311 0.221 −0.147 0.322 1.128
theta_hat_t_stats 100 0 1.407 0.889 −0.554 1.470 3.165
R2_demand 100 0 0.377 0.113 0.060 0.406 0.605
R2_supply 100 0 0.972 0.005 0.952 0.972 0.980
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.009 0.199 9.552 10.022 10.515
alpha1_hat 100 0 1.064 0.807 −1.336 1.050 3.026
alpha2_hat 100 0 0.997 0.092 0.774 0.993 1.273
alpha3_hat 100 0 1.006 0.052 0.901 1.006 1.140
gamma0_hat 100 0 1.004 0.274 0.250 0.994 1.552
gamma1_hat 100 0 0.893 0.917 −1.210 0.904 2.858
gamma2_hat 100 0 0.999 0.039 0.931 0.992 1.100
gamma3_hat 100 0 1.004 0.042 0.910 1.003 1.102
theta_hat 100 0 0.339 0.102 0.130 0.340 0.599
theta_hat_t_stats 100 0 3.622 0.928 1.590 3.607 5.669
R2_demand 100 0 0.392 0.048 0.264 0.393 0.500
R2_supply 100 0 0.972 0.002 0.964 0.972 0.978
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.983 0.130 9.671 9.990 10.252
alpha1_hat 100 0 1.070 0.629 −0.608 1.069 2.884
alpha2_hat 100 0 0.989 0.064 0.812 0.983 1.179
alpha3_hat 100 0 0.996 0.032 0.915 0.996 1.087
gamma0_hat 100 0 0.978 0.229 0.326 0.977 1.440
gamma1_hat 100 0 1.009 0.640 −0.480 1.067 2.528
gamma2_hat 100 0 1.002 0.034 0.939 1.000 1.092
gamma3_hat 100 0 1.002 0.038 0.935 0.997 1.104
theta_hat 100 0 0.335 0.065 0.178 0.330 0.476
theta_hat_t_stats 100 0 5.070 0.843 2.943 4.967 7.079
R2_demand 100 0 0.400 0.034 0.302 0.401 0.492
R2_supply 100 0 0.971 0.002 0.966 0.972 0.975
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.015 0.085 9.826 10.024 10.269
alpha1_hat 100 0 1.057 0.391 −0.222 1.003 2.088
alpha2_hat 100 0 0.997 0.043 0.904 0.998 1.133
alpha3_hat 100 0 1.003 0.020 0.961 1.002 1.064
gamma0_hat 100 0 1.005 0.131 0.648 1.014 1.380
gamma1_hat 100 0 0.984 0.408 0.060 0.983 1.755
gamma2_hat 100 0 0.999 0.019 0.950 1.000 1.047
gamma3_hat 100 0 1.000 0.023 0.946 1.001 1.059
theta_hat 100 0 0.330 0.040 0.241 0.331 0.426
theta_hat_t_stats 100 0 8.052 0.916 5.756 8.065 10.312
R2_demand 100 0 0.395 0.020 0.347 0.395 0.445
R2_supply 100 0 0.972 0.001 0.969 0.972 0.974

\(\alpha_2 = 5.0\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.155 1.110 8.022 9.940 14.074
alpha1_hat 100 0 −0.743 17.362 −47.100 −1.777 52.841
alpha2_hat 100 0 5.352 1.980 0.878 5.258 10.056
alpha3_hat 100 0 1.028 0.251 0.492 0.996 2.140
gamma0_hat 100 0 0.539 2.743 −15.815 1.036 3.283
gamma1_hat 100 0 2.477 22.916 −105.349 3.018 69.242
gamma2_hat 100 0 1.036 0.375 0.531 0.967 3.254
gamma3_hat 100 0 1.076 0.435 0.566 1.011 4.192
theta_hat 100 0 0.424 0.697 −0.752 0.225 4.011
theta_hat_t_stats 100 0 0.650 0.790 −1.640 0.623 2.371
R2_demand 100 0 0.344 0.381 −1.444 0.452 0.757
R2_supply 100 0 0.965 0.034 0.738 0.972 0.991
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.034 0.656 8.761 9.962 12.255
alpha1_hat 100 0 1.650 12.396 −33.550 0.718 38.354
alpha2_hat 100 0 4.988 1.422 2.300 4.944 9.145
alpha3_hat 100 0 1.001 0.156 0.577 1.020 1.546
gamma0_hat 100 0 0.989 1.015 −1.669 1.226 2.975
gamma1_hat 100 0 −1.129 15.342 −40.129 −0.687 35.773
gamma2_hat 100 0 1.015 0.161 0.705 1.003 1.458
gamma3_hat 100 0 0.982 0.136 0.668 0.976 1.400
theta_hat 100 0 0.368 0.330 −0.324 0.357 1.578
theta_hat_t_stats 100 0 1.177 0.895 −1.420 1.283 2.479
R2_demand 100 0 0.424 0.158 −0.076 0.453 0.721
R2_supply 100 0 0.970 0.008 0.945 0.971 0.985
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.080 0.461 9.105 10.044 11.438
alpha1_hat 100 0 2.168 8.411 −18.957 2.191 21.653
alpha2_hat 100 0 4.968 0.909 2.360 4.973 7.329
alpha3_hat 100 0 1.003 0.094 0.807 0.992 1.269
gamma0_hat 100 0 0.991 0.655 −0.938 1.083 2.279
gamma1_hat 100 0 −1.505 9.800 −34.495 −1.450 22.510
gamma2_hat 100 0 0.996 0.101 0.776 0.982 1.237
gamma3_hat 100 0 1.000 0.100 0.813 0.998 1.248
theta_hat 100 0 0.381 0.241 −0.110 0.336 1.250
theta_hat_t_stats 100 0 1.849 0.899 −0.590 1.943 4.052
R2_demand 100 0 0.424 0.107 0.069 0.441 0.604
R2_supply 100 0 0.970 0.005 0.949 0.970 0.979
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.017 0.207 9.502 10.010 10.834
alpha1_hat 100 0 1.106 3.724 −11.821 0.780 11.378
alpha2_hat 100 0 5.009 0.401 4.036 5.032 6.438
alpha3_hat 100 0 0.998 0.050 0.890 1.000 1.130
gamma0_hat 100 0 0.964 0.330 0.185 0.961 1.600
gamma1_hat 100 0 0.862 4.285 −10.247 1.234 9.269
gamma2_hat 100 0 1.004 0.052 0.881 0.999 1.128
gamma3_hat 100 0 1.000 0.045 0.885 1.000 1.095
theta_hat 100 0 0.338 0.095 0.096 0.328 0.580
theta_hat_t_stats 100 0 3.998 0.955 1.308 4.038 6.275
R2_demand 100 0 0.435 0.049 0.297 0.443 0.561
R2_supply 100 0 0.969 0.003 0.959 0.969 0.974
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.998 0.146 9.661 10.001 10.377
alpha1_hat 100 0 0.880 2.928 −9.563 0.832 8.123
alpha2_hat 100 0 5.011 0.310 4.243 5.014 6.091
alpha3_hat 100 0 0.998 0.030 0.908 0.999 1.086
gamma0_hat 100 0 1.014 0.230 0.491 1.038 1.523
gamma1_hat 100 0 0.887 2.866 −7.982 0.992 7.093
gamma2_hat 100 0 0.999 0.034 0.902 1.004 1.083
gamma3_hat 100 0 0.999 0.032 0.934 0.995 1.096
theta_hat 100 0 0.330 0.069 0.186 0.322 0.474
theta_hat_t_stats 100 0 5.620 0.949 3.367 5.567 7.275
R2_demand 100 0 0.437 0.032 0.337 0.438 0.501
R2_supply 100 0 0.969 0.002 0.965 0.970 0.974
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.998 0.088 9.713 9.995 10.212
alpha1_hat 100 0 0.987 1.683 −3.239 0.900 5.071
alpha2_hat 100 0 4.998 0.194 4.540 4.998 5.521
alpha3_hat 100 0 1.001 0.020 0.962 1.003 1.053
gamma0_hat 100 0 1.017 0.154 0.674 1.008 1.517
gamma1_hat 100 0 1.055 1.585 −2.988 1.077 5.520
gamma2_hat 100 0 0.999 0.021 0.934 1.001 1.047
gamma3_hat 100 0 0.998 0.022 0.948 0.997 1.055
theta_hat 100 0 0.326 0.035 0.222 0.328 0.427
theta_hat_t_stats 100 0 8.812 0.773 6.410 8.909 10.602
R2_demand 100 0 0.439 0.023 0.391 0.439 0.495
R2_supply 100 0 0.969 0.001 0.966 0.969 0.972

\(\alpha_2 = 20.0\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.012 1.402 7.889 9.766 20.673
alpha1_hat 100 0 7.425 85.038 −130.833 7.690 601.521
alpha2_hat 100 0 19.400 7.157 3.583 18.262 41.696
alpha3_hat 100 0 1.006 0.267 0.477 0.984 2.735
gamma0_hat 100 0 0.855 1.443 −3.955 0.769 4.436
gamma1_hat 100 0 −27.478 100.281 −423.172 −8.324 164.203
gamma2_hat 100 0 1.021 0.218 0.473 1.035 1.712
gamma3_hat 100 0 1.045 0.215 0.485 1.019 1.618
theta_hat 100 0 0.507 0.610 −0.570 0.374 3.227
theta_hat_t_stats 100 0 0.880 0.818 −1.123 0.917 2.508
R2_demand 100 0 0.371 0.875 −7.955 0.533 0.791
R2_supply 100 0 0.969 0.010 0.932 0.971 0.987
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.079 0.710 8.357 9.956 13.374
alpha1_hat 100 0 −5.827 50.208 −119.702 1.818 123.228
alpha2_hat 100 0 20.975 6.353 4.808 19.588 36.780
alpha3_hat 100 0 1.019 0.150 0.700 0.997 1.424
gamma0_hat 100 0 0.844 1.080 −2.479 0.841 2.832
gamma1_hat 100 0 −1.076 73.058 −309.002 5.744 216.125
gamma2_hat 100 0 1.035 0.164 0.622 1.004 1.578
gamma3_hat 100 0 1.001 0.159 0.653 0.989 1.507
theta_hat 100 0 0.370 0.455 −0.590 0.262 2.811
theta_hat_t_stats 100 0 1.149 0.941 −1.432 1.189 4.282
R2_demand 100 0 0.415 0.203 −0.875 0.450 0.720
R2_supply 100 0 0.969 0.008 0.941 0.971 0.982
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.922 0.407 9.079 9.935 10.884
alpha1_hat 100 0 −1.508 28.924 −68.330 −3.907 68.355
alpha2_hat 100 0 19.881 3.396 11.564 19.740 28.292
alpha3_hat 100 0 0.982 0.101 0.768 0.984 1.289
gamma0_hat 100 0 0.963 0.740 −1.027 1.044 2.517
gamma1_hat 100 0 3.467 39.149 −95.363 5.773 124.265
gamma2_hat 100 0 1.001 0.107 0.722 0.993 1.323
gamma3_hat 100 0 1.007 0.115 0.692 1.011 1.284
theta_hat 100 0 0.335 0.224 −0.319 0.307 1.260
theta_hat_t_stats 100 0 1.666 0.957 −1.529 1.725 3.495
R2_demand 100 0 0.446 0.109 0.103 0.464 0.666
R2_supply 100 0 0.969 0.006 0.954 0.970 0.978
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.014 0.213 9.553 10.005 10.726
alpha1_hat 100 0 −0.823 13.567 −28.407 −2.822 40.858
alpha2_hat 100 0 20.238 1.701 16.009 20.199 24.180
alpha3_hat 100 0 1.003 0.048 0.920 1.002 1.161
gamma0_hat 100 0 0.996 0.287 0.179 1.020 1.546
gamma1_hat 100 0 0.838 15.014 −52.139 2.099 33.161
gamma2_hat 100 0 1.000 0.046 0.914 1.001 1.111
gamma3_hat 100 0 1.009 0.043 0.914 1.004 1.135
theta_hat 100 0 0.332 0.085 0.141 0.328 0.651
theta_hat_t_stats 100 0 4.049 0.869 2.059 3.945 6.470
R2_demand 100 0 0.450 0.053 0.302 0.458 0.542
R2_supply 100 0 0.969 0.002 0.962 0.969 0.973
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.988 0.137 9.717 9.973 10.406
alpha1_hat 100 0 1.148 10.368 −31.475 0.356 30.391
alpha2_hat 100 0 19.959 1.206 17.236 19.892 23.249
alpha3_hat 100 0 0.999 0.031 0.925 0.999 1.086
gamma0_hat 100 0 0.963 0.199 0.531 0.942 1.557
gamma1_hat 100 0 1.428 9.773 −22.716 1.421 22.038
gamma2_hat 100 0 1.004 0.028 0.907 1.002 1.069
gamma3_hat 100 0 1.001 0.033 0.920 1.001 1.070
theta_hat 100 0 0.334 0.057 0.212 0.335 0.480
theta_hat_t_stats 100 0 5.750 0.827 4.022 5.721 7.626
R2_demand 100 0 0.446 0.035 0.350 0.447 0.527
R2_supply 100 0 0.969 0.002 0.965 0.969 0.973
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.001 0.098 9.752 10.008 10.213
alpha1_hat 100 0 1.032 6.687 −12.985 1.126 13.148
alpha2_hat 100 0 20.003 0.827 17.757 19.938 22.084
alpha3_hat 100 0 1.001 0.022 0.938 1.002 1.048
gamma0_hat 100 0 1.011 0.140 0.692 1.023 1.314
gamma1_hat 100 0 0.305 6.786 −21.431 1.053 16.437
gamma2_hat 100 0 0.999 0.021 0.960 1.000 1.050
gamma3_hat 100 0 1.000 0.020 0.955 1.000 1.046
theta_hat 100 0 0.332 0.042 0.211 0.330 0.483
theta_hat_t_stats 100 0 9.119 0.910 6.381 9.119 12.007
R2_demand 100 0 0.447 0.022 0.389 0.449 0.490
R2_supply 100 0 0.969 0.001 0.966 0.969 0.971

\(\theta = 0.5\)

\(\alpha_2 = 0.1\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.068 1.045 7.915 9.982 13.936
alpha1_hat 100 0 1.002 1.373 −1.912 1.049 7.658
alpha2_hat 100 0 0.104 0.129 −0.276 0.093 0.585
alpha3_hat 100 0 0.987 0.208 0.523 0.976 1.566
gamma0_hat 100 0 0.738 2.124 −9.781 1.169 3.994
gamma1_hat 100 0 3.745 27.350 −15.803 1.268 265.584
gamma2_hat 100 0 1.013 0.253 0.454 0.956 1.860
gamma3_hat 100 0 1.032 0.233 0.498 1.021 1.832
theta_hat 100 0 −0.371 11.123 −104.077 0.420 13.239
theta_hat_t_stats 100 0 0.380 1.033 −2.757 0.571 2.346
R2_demand 100 0 0.125 0.446 −2.641 0.237 0.556
R2_supply 100 0 0.978 0.020 0.828 0.983 0.992
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.942 0.620 8.837 9.822 12.284
alpha1_hat 100 0 1.077 0.863 −0.684 1.102 5.283
alpha2_hat 100 0 0.089 0.076 −0.161 0.093 0.261
alpha3_hat 100 0 0.994 0.148 0.680 0.990 1.396
gamma0_hat 100 0 0.998 1.121 −2.465 1.236 2.971
gamma1_hat 100 0 0.563 4.488 −26.166 0.846 17.330
gamma2_hat 100 0 0.996 0.170 0.620 1.001 1.466
gamma3_hat 100 0 1.008 0.139 0.745 1.000 1.398
theta_hat 100 0 0.757 2.486 −11.887 0.633 14.242
theta_hat_t_stats 100 0 0.679 1.047 −2.177 0.649 3.384
R2_demand 100 0 0.200 0.209 −0.678 0.235 0.613
R2_supply 100 0 0.980 0.006 0.958 0.982 0.987
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.972 0.468 8.912 9.949 10.966
alpha1_hat 100 0 0.955 0.510 −0.215 0.908 2.705
alpha2_hat 100 0 0.102 0.050 −0.020 0.098 0.219
alpha3_hat 100 0 0.983 0.101 0.773 0.981 1.267
gamma0_hat 100 0 0.951 0.743 −2.020 1.069 2.575
gamma1_hat 100 0 0.952 2.608 −10.691 0.876 19.690
gamma2_hat 100 0 1.006 0.110 0.716 0.992 1.472
gamma3_hat 100 0 1.007 0.098 0.798 1.009 1.298
theta_hat 100 0 0.560 1.561 −10.086 0.565 9.036
theta_hat_t_stats 100 0 1.021 1.023 −2.788 1.057 3.733
R2_demand 100 0 0.188 0.151 −0.202 0.204 0.447
R2_supply 100 0 0.980 0.005 0.959 0.980 0.988
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.992 0.197 9.598 9.982 10.443
alpha1_hat 100 0 0.975 0.235 0.360 0.962 1.452
alpha2_hat 100 0 0.102 0.023 0.059 0.100 0.187
alpha3_hat 100 0 1.001 0.046 0.915 0.997 1.101
gamma0_hat 100 0 1.039 0.304 0.188 1.046 1.789
gamma1_hat 100 0 1.062 0.491 −0.258 1.052 2.196
gamma2_hat 100 0 0.997 0.045 0.897 0.998 1.134
gamma3_hat 100 0 0.993 0.049 0.880 0.995 1.139
theta_hat 100 0 0.462 0.262 −0.104 0.443 1.152
theta_hat_t_stats 100 0 2.045 1.003 −0.543 2.065 3.956
R2_demand 100 0 0.195 0.066 0.021 0.200 0.320
R2_supply 100 0 0.980 0.001 0.976 0.980 0.983
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.008 0.139 9.666 10.013 10.278
alpha1_hat 100 0 0.995 0.176 0.583 0.985 1.451
alpha2_hat 100 0 0.101 0.017 0.064 0.102 0.144
alpha3_hat 100 0 1.005 0.030 0.930 1.005 1.080
gamma0_hat 100 0 0.971 0.218 0.494 0.981 1.460
gamma1_hat 100 0 0.942 0.341 −0.036 0.998 1.505
gamma2_hat 100 0 1.004 0.034 0.932 1.003 1.078
gamma3_hat 100 0 1.001 0.031 0.933 1.000 1.105
theta_hat 100 0 0.533 0.171 0.211 0.507 0.969
theta_hat_t_stats 100 0 3.396 0.832 1.488 3.371 5.278
R2_demand 100 0 0.202 0.041 0.083 0.205 0.277
R2_supply 100 0 0.980 0.001 0.977 0.980 0.982
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.005 0.084 9.802 10.012 10.204
alpha1_hat 100 0 1.022 0.106 0.801 1.032 1.255
alpha2_hat 100 0 0.098 0.010 0.077 0.098 0.123
alpha3_hat 100 0 1.001 0.021 0.945 0.999 1.046
gamma0_hat 100 0 1.004 0.140 0.666 1.009 1.300
gamma1_hat 100 0 0.980 0.233 0.471 0.961 1.494
gamma2_hat 100 0 0.999 0.020 0.959 1.000 1.057
gamma3_hat 100 0 0.997 0.020 0.951 0.996 1.051
theta_hat 100 0 0.507 0.117 0.244 0.514 0.770
theta_hat_t_stats 100 0 5.105 0.987 2.844 5.194 7.343
R2_demand 100 0 0.200 0.028 0.140 0.198 0.277
R2_supply 100 0 0.980 0.001 0.978 0.980 0.982

\(\alpha_2 = 0.5\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.918 1.030 7.960 9.700 14.079
alpha1_hat 100 0 0.635 2.503 −4.577 0.383 7.811
alpha2_hat 100 0 0.522 0.295 −0.166 0.537 1.309
alpha3_hat 100 0 0.993 0.217 0.623 0.971 1.780
gamma0_hat 100 0 0.330 1.867 −7.538 0.554 3.598
gamma1_hat 100 0 −3.952 43.802 −412.503 1.299 40.828
gamma2_hat 100 0 1.066 0.258 0.526 1.025 1.782
gamma3_hat 100 0 1.056 0.285 0.593 0.997 2.381
theta_hat 100 0 1.763 9.830 −6.911 0.527 95.646
theta_hat_t_stats 100 0 0.822 0.981 −2.637 0.863 3.001
R2_demand 100 0 0.262 0.375 −1.820 0.348 0.696
R2_supply 100 0 0.973 0.014 0.902 0.976 0.993
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.042 0.710 8.658 9.920 12.798
alpha1_hat 100 0 1.142 1.940 −3.704 0.923 7.108
alpha2_hat 100 0 0.494 0.197 −0.054 0.486 1.028
alpha3_hat 100 0 1.019 0.154 0.644 1.032 1.481
gamma0_hat 100 0 1.102 1.004 −1.346 1.185 3.312
gamma1_hat 100 0 0.687 6.103 −21.565 1.301 41.843
gamma2_hat 100 0 0.993 0.147 0.644 0.981 1.395
gamma3_hat 100 0 0.989 0.140 0.699 1.000 1.514
theta_hat 100 0 0.549 1.016 −5.737 0.387 3.951
theta_hat_t_stats 100 0 1.127 0.960 −1.718 1.152 2.974
R2_demand 100 0 0.260 0.243 −1.117 0.336 0.554
R2_supply 100 0 0.977 0.006 0.956 0.978 0.987
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.008 0.452 9.236 9.951 11.002
alpha1_hat 100 0 0.901 1.288 −2.563 0.945 4.009
alpha2_hat 100 0 0.511 0.126 0.169 0.517 0.786
alpha3_hat 100 0 0.992 0.093 0.735 0.992 1.356
gamma0_hat 100 0 1.006 0.654 −0.650 0.996 2.533
gamma1_hat 100 0 0.888 1.891 −8.429 0.982 6.502
gamma2_hat 100 0 0.997 0.101 0.789 0.992 1.328
gamma3_hat 100 0 0.993 0.107 0.778 0.981 1.295
theta_hat 100 0 0.526 0.355 −0.233 0.480 2.127
theta_hat_t_stats 100 0 1.860 0.876 −0.426 1.991 3.695
R2_demand 100 0 0.270 0.137 −0.168 0.277 0.526
R2_supply 100 0 0.976 0.004 0.965 0.977 0.985
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.985 0.190 9.511 9.980 10.607
alpha1_hat 100 0 1.057 0.577 −0.562 1.106 2.106
alpha2_hat 100 0 0.492 0.063 0.370 0.482 0.652
alpha3_hat 100 0 0.997 0.045 0.884 0.996 1.102
gamma0_hat 100 0 0.982 0.338 −0.035 1.001 2.124
gamma1_hat 100 0 0.902 0.769 −0.901 0.913 2.540
gamma2_hat 100 0 1.001 0.050 0.801 1.000 1.140
gamma3_hat 100 0 1.003 0.046 0.873 0.999 1.142
theta_hat 100 0 0.524 0.125 0.268 0.531 0.880
theta_hat_t_stats 100 0 4.388 0.892 2.378 4.412 6.896
R2_demand 100 0 0.292 0.056 0.164 0.300 0.415
R2_supply 100 0 0.976 0.002 0.971 0.976 0.981
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.997 0.154 9.672 9.988 10.373
alpha1_hat 100 0 0.951 0.370 0.116 0.988 2.276
alpha2_hat 100 0 0.505 0.040 0.409 0.505 0.618
alpha3_hat 100 0 1.001 0.034 0.914 0.997 1.070
gamma0_hat 100 0 0.964 0.232 0.407 0.968 1.403
gamma1_hat 100 0 1.000 0.510 −0.121 0.996 2.040
gamma2_hat 100 0 1.005 0.035 0.936 1.008 1.084
gamma3_hat 100 0 1.004 0.036 0.907 1.003 1.092
theta_hat 100 0 0.507 0.091 0.315 0.521 0.694
theta_hat_t_stats 100 0 6.240 0.928 4.305 6.243 8.591
R2_demand 100 0 0.286 0.046 0.149 0.289 0.386
R2_supply 100 0 0.976 0.001 0.973 0.976 0.979
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.001 0.085 9.719 10.004 10.156
alpha1_hat 100 0 1.011 0.213 0.549 1.025 1.559
alpha2_hat 100 0 0.499 0.024 0.432 0.501 0.555
alpha3_hat 100 0 0.998 0.019 0.940 0.999 1.048
gamma0_hat 100 0 0.994 0.146 0.514 0.997 1.269
gamma1_hat 100 0 1.053 0.346 −0.060 1.089 1.882
gamma2_hat 100 0 1.001 0.022 0.957 1.002 1.054
gamma3_hat 100 0 1.001 0.021 0.940 0.997 1.069
theta_hat 100 0 0.491 0.065 0.342 0.489 0.675
theta_hat_t_stats 100 0 9.521 1.047 6.917 9.426 12.705
R2_demand 100 0 0.287 0.024 0.233 0.286 0.368
R2_supply 100 0 0.976 0.001 0.974 0.976 0.978

\(\alpha_2 = 1.0\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.025 1.054 8.416 9.819 13.799
alpha1_hat 100 0 1.174 4.966 −12.688 0.825 25.156
alpha2_hat 100 0 0.989 0.588 −0.632 0.949 2.952
alpha3_hat 100 0 1.004 0.258 0.489 0.952 1.923
gamma0_hat 100 0 0.814 1.716 −9.207 0.869 4.640
gamma1_hat 100 0 −3.798 28.528 −259.759 2.076 17.332
gamma2_hat 100 0 1.034 0.225 0.592 1.059 2.011
gamma3_hat 100 0 1.009 0.230 0.473 1.016 1.900
theta_hat 100 0 1.090 3.228 −1.121 0.344 30.113
theta_hat_t_stats 100 0 0.864 0.940 −1.943 0.946 3.401
R2_demand 100 0 0.275 0.424 −2.122 0.397 0.744
R2_supply 100 0 0.975 0.015 0.859 0.978 0.991
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.014 0.683 8.583 9.864 12.234
alpha1_hat 100 0 0.899 2.964 −7.428 0.718 8.488
alpha2_hat 100 0 1.017 0.314 0.322 1.005 2.141
alpha3_hat 100 0 0.993 0.136 0.714 0.979 1.387
gamma0_hat 100 0 0.880 0.947 −2.870 0.853 2.937
gamma1_hat 100 0 0.723 4.365 −14.154 1.815 9.577
gamma2_hat 100 0 1.019 0.142 0.696 1.016 1.576
gamma3_hat 100 0 1.008 0.136 0.677 0.994 1.416
theta_hat 100 0 0.542 0.431 −0.306 0.429 2.409
theta_hat_t_stats 100 0 1.448 0.846 −0.844 1.372 3.426
R2_demand 100 0 0.304 0.191 −0.249 0.329 0.608
R2_supply 100 0 0.976 0.006 0.954 0.976 0.986
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.052 0.488 8.847 10.022 11.306
alpha1_hat 100 0 1.198 2.177 −3.931 1.367 6.972
alpha2_hat 100 0 0.993 0.241 0.512 0.976 1.761
alpha3_hat 100 0 1.016 0.106 0.780 1.002 1.317
gamma0_hat 100 0 0.862 0.718 −2.439 0.947 2.256
gamma1_hat 100 0 0.604 2.878 −7.175 1.118 6.459
gamma2_hat 100 0 1.011 0.102 0.800 0.993 1.481
gamma3_hat 100 0 1.015 0.098 0.820 0.999 1.366
theta_hat 100 0 0.571 0.309 −0.035 0.516 1.487
theta_hat_t_stats 100 0 2.176 0.873 −0.160 2.116 4.171
R2_demand 100 0 0.293 0.138 −0.167 0.305 0.528
R2_supply 100 0 0.975 0.005 0.955 0.976 0.983
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.032 0.204 9.599 10.041 10.669
alpha1_hat 100 0 1.042 0.942 −1.293 1.059 3.488
alpha2_hat 100 0 1.005 0.111 0.657 1.021 1.303
alpha3_hat 100 0 1.004 0.041 0.910 1.006 1.118
gamma0_hat 100 0 0.930 0.343 0.017 0.994 1.878
gamma1_hat 100 0 1.012 1.121 −1.948 1.030 4.781
gamma2_hat 100 0 1.006 0.052 0.900 0.999 1.138
gamma3_hat 100 0 1.007 0.041 0.875 1.007 1.106
theta_hat 100 0 0.510 0.120 0.157 0.505 0.916
theta_hat_t_stats 100 0 4.850 0.867 1.761 4.874 6.715
R2_demand 100 0 0.313 0.053 0.145 0.312 0.413
R2_supply 100 0 0.975 0.002 0.968 0.975 0.980
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.032 0.154 9.640 10.025 10.500
alpha1_hat 100 0 1.070 0.659 −0.595 0.976 2.432
alpha2_hat 100 0 1.003 0.079 0.770 1.007 1.191
alpha3_hat 100 0 1.005 0.031 0.933 1.000 1.081
gamma0_hat 100 0 0.959 0.243 0.362 0.964 1.572
gamma1_hat 100 0 1.038 0.850 −1.009 1.011 3.044
gamma2_hat 100 0 1.008 0.034 0.924 1.006 1.079
gamma3_hat 100 0 1.004 0.031 0.916 1.004 1.088
theta_hat 100 0 0.498 0.081 0.262 0.494 0.674
theta_hat_t_stats 100 0 6.805 0.913 4.385 6.800 9.071
R2_demand 100 0 0.313 0.042 0.214 0.318 0.400
R2_supply 100 0 0.975 0.002 0.971 0.975 0.980
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.987 0.095 9.744 9.980 10.205
alpha1_hat 100 0 0.971 0.444 −0.180 0.945 2.023
alpha2_hat 100 0 0.998 0.049 0.888 0.999 1.115
alpha3_hat 100 0 0.996 0.021 0.948 0.993 1.054
gamma0_hat 100 0 1.017 0.139 0.709 0.993 1.369
gamma1_hat 100 0 1.077 0.439 0.033 1.085 2.112
gamma2_hat 100 0 0.997 0.021 0.945 0.999 1.064
gamma3_hat 100 0 0.997 0.021 0.949 0.996 1.056
theta_hat 100 0 0.494 0.045 0.372 0.495 0.612
theta_hat_t_stats 100 0 10.736 0.764 8.892 10.741 12.523
R2_demand 100 0 0.320 0.025 0.255 0.319 0.366
R2_supply 100 0 0.975 0.001 0.973 0.975 0.977

\(\alpha_2 = 5.0\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.905 0.857 7.903 9.839 13.046
alpha1_hat 100 0 1.229 19.905 −56.477 −1.308 46.467
alpha2_hat 100 0 4.838 2.291 −0.843 5.057 11.967
alpha3_hat 100 0 0.994 0.178 0.653 0.978 1.503
gamma0_hat 100 0 0.737 1.491 −4.036 1.038 3.492
gamma1_hat 100 0 −5.312 160.588 −817.729 −0.157 1227.512
gamma2_hat 100 0 1.031 0.235 0.594 0.993 1.703
gamma3_hat 100 0 1.046 0.223 0.616 1.027 1.849
theta_hat 100 0 0.649 4.575 −36.293 0.580 20.544
theta_hat_t_stats 100 0 1.087 0.931 −2.335 1.315 2.911
R2_demand 100 0 0.355 0.287 −1.042 0.408 0.736
R2_supply 100 0 0.974 0.010 0.938 0.977 0.988
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.060 0.675 8.425 10.004 12.044
alpha1_hat 100 0 0.282 13.587 −33.200 −1.095 43.051
alpha2_hat 100 0 5.147 1.474 0.676 5.129 9.030
alpha3_hat 100 0 0.999 0.156 0.745 0.993 1.553
gamma0_hat 100 0 0.844 1.140 −2.899 1.024 2.943
gamma1_hat 100 0 0.865 26.412 −216.314 3.537 34.904
gamma2_hat 100 0 1.012 0.164 0.663 0.998 1.461
gamma3_hat 100 0 1.014 0.162 0.692 1.028 1.501
theta_hat 100 0 0.549 0.553 −0.152 0.444 4.770
theta_hat_t_stats 100 0 1.443 0.743 −0.614 1.499 2.981
R2_demand 100 0 0.312 0.260 −0.988 0.370 0.586
R2_supply 100 0 0.973 0.008 0.933 0.974 0.984
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.960 0.473 8.789 9.966 11.243
alpha1_hat 100 0 −0.095 9.647 −19.686 0.279 20.684
alpha2_hat 100 0 5.058 1.106 2.841 4.852 8.631
alpha3_hat 100 0 0.994 0.114 0.770 0.985 1.251
gamma0_hat 100 0 0.880 0.783 −2.403 0.970 2.485
gamma1_hat 100 0 0.108 12.032 −39.482 0.970 26.673
gamma2_hat 100 0 1.010 0.117 0.754 1.012 1.428
gamma3_hat 100 0 1.014 0.124 0.776 0.992 1.414
theta_hat 100 0 0.552 0.268 −0.004 0.518 1.557
theta_hat_t_stats 100 0 2.441 0.876 −0.022 2.376 4.325
R2_demand 100 0 0.342 0.130 −0.090 0.361 0.561
R2_supply 100 0 0.973 0.005 0.950 0.974 0.984
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.016 0.184 9.555 10.003 10.435
alpha1_hat 100 0 1.571 3.895 −7.121 1.534 10.478
alpha2_hat 100 0 4.965 0.453 3.772 4.948 5.792
alpha3_hat 100 0 1.003 0.043 0.913 1.002 1.110
gamma0_hat 100 0 1.013 0.281 0.181 0.998 1.616
gamma1_hat 100 0 0.285 4.807 −9.959 0.197 11.288
gamma2_hat 100 0 0.999 0.040 0.905 1.003 1.133
gamma3_hat 100 0 0.994 0.044 0.886 0.990 1.122
theta_hat 100 0 0.511 0.099 0.243 0.515 0.698
theta_hat_t_stats 100 0 5.378 0.876 3.201 5.441 7.254
R2_demand 100 0 0.341 0.053 0.182 0.347 0.431
R2_supply 100 0 0.974 0.002 0.969 0.974 0.978
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.005 0.139 9.680 10.007 10.451
alpha1_hat 100 0 1.166 2.547 −3.731 0.902 6.745
alpha2_hat 100 0 4.994 0.341 4.223 5.031 5.811
alpha3_hat 100 0 1.004 0.034 0.927 1.005 1.089
gamma0_hat 100 0 1.003 0.201 0.508 1.017 1.605
gamma1_hat 100 0 0.523 3.522 −11.355 0.797 9.824
gamma2_hat 100 0 0.999 0.028 0.905 0.997 1.072
gamma3_hat 100 0 0.998 0.029 0.883 1.000 1.047
theta_hat 100 0 0.509 0.084 0.293 0.506 0.722
theta_hat_t_stats 100 0 7.599 0.987 4.524 7.715 10.192
R2_demand 100 0 0.344 0.042 0.237 0.346 0.452
R2_supply 100 0 0.974 0.001 0.970 0.974 0.976
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.000 0.085 9.800 10.005 10.220
alpha1_hat 100 0 0.678 1.526 −2.974 0.854 3.926
alpha2_hat 100 0 5.032 0.195 4.544 5.025 5.640
alpha3_hat 100 0 0.998 0.018 0.954 1.001 1.045
gamma0_hat 100 0 0.986 0.142 0.638 0.985 1.294
gamma1_hat 100 0 1.516 1.977 −3.769 1.449 6.051
gamma2_hat 100 0 1.001 0.021 0.952 1.001 1.053
gamma3_hat 100 0 1.001 0.019 0.962 0.998 1.050
theta_hat 100 0 0.492 0.046 0.384 0.492 0.630
theta_hat_t_stats 100 0 11.836 0.895 10.008 11.875 13.898
R2_demand 100 0 0.348 0.023 0.293 0.348 0.397
R2_supply 100 0 0.974 0.001 0.971 0.974 0.976

\(\alpha_2 = 20.0\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.952 1.012 8.115 9.864 13.583
alpha1_hat 100 0 −6.081 81.140 −175.858 −5.806 224.524
alpha2_hat 100 0 20.447 8.570 −2.178 20.765 39.275
alpha3_hat 100 0 0.966 0.235 0.550 0.939 1.762
gamma0_hat 100 0 0.839 1.760 −7.683 1.029 4.745
gamma1_hat 100 0 −209.454 1634.353 −16118.530 −2.292 213.426
gamma2_hat 100 0 1.024 0.252 0.573 1.000 2.014
gamma3_hat 100 0 1.041 0.246 0.507 1.020 1.782
theta_hat 100 0 1.795 9.343 −0.650 0.528 92.086
theta_hat_t_stats 100 0 1.147 0.959 −1.824 1.186 3.131
R2_demand 100 0 0.319 0.389 −1.852 0.434 0.806
R2_supply 100 0 0.973 0.018 0.848 0.977 0.990
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.919 0.624 8.551 9.892 12.815
alpha1_hat 100 0 3.180 50.544 −105.249 3.688 130.740
alpha2_hat 100 0 19.360 6.359 5.189 18.479 43.643
alpha3_hat 100 0 1.000 0.152 0.649 0.982 1.449
gamma0_hat 100 0 1.025 0.794 −0.906 1.065 3.289
gamma1_hat 100 0 −13.671 68.100 −233.919 −0.987 132.235
gamma2_hat 100 0 1.000 0.125 0.665 0.995 1.288
gamma3_hat 100 0 0.995 0.142 0.656 0.976 1.316
theta_hat 100 0 0.599 0.411 −0.114 0.550 2.040
theta_hat_t_stats 100 0 1.672 0.864 −0.411 1.635 3.555
R2_demand 100 0 0.351 0.274 −1.872 0.380 0.676
R2_supply 100 0 0.974 0.006 0.961 0.975 0.986
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.013 0.473 8.633 9.987 11.164
alpha1_hat 100 0 0.905 38.348 −127.127 7.208 71.531
alpha2_hat 100 0 20.126 4.806 9.769 19.467 38.683
alpha3_hat 100 0 1.013 0.100 0.800 1.018 1.314
gamma0_hat 100 0 0.966 0.730 −2.368 1.007 2.269
gamma1_hat 100 0 −5.638 39.944 −115.338 0.932 78.109
gamma2_hat 100 0 0.995 0.109 0.738 0.993 1.296
gamma3_hat 100 0 1.007 0.112 0.767 1.010 1.450
theta_hat 100 0 0.557 0.275 0.022 0.550 1.717
theta_hat_t_stats 100 0 2.454 0.875 0.119 2.507 4.810
R2_demand 100 0 0.326 0.140 −0.064 0.359 0.547
R2_supply 100 0 0.974 0.005 0.950 0.975 0.982
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.992 0.192 9.555 9.974 10.513
alpha1_hat 100 0 2.092 14.772 −39.193 3.266 45.123
alpha2_hat 100 0 19.835 1.710 14.689 19.769 23.925
alpha3_hat 100 0 1.002 0.042 0.916 0.996 1.100
gamma0_hat 100 0 1.012 0.291 0.075 0.997 1.996
gamma1_hat 100 0 2.384 16.937 −40.210 2.903 41.254
gamma2_hat 100 0 0.997 0.044 0.879 1.000 1.134
gamma3_hat 100 0 1.004 0.048 0.911 1.003 1.147
theta_hat 100 0 0.494 0.096 0.264 0.503 0.706
theta_hat_t_stats 100 0 5.299 0.847 3.097 5.434 7.016
R2_demand 100 0 0.356 0.048 0.230 0.362 0.464
R2_supply 100 0 0.974 0.002 0.968 0.973 0.978
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.992 0.192 9.555 9.974 10.513
alpha1_hat 100 0 2.092 14.772 −39.193 3.266 45.123
alpha2_hat 100 0 19.835 1.710 14.689 19.769 23.925
alpha3_hat 100 0 1.002 0.042 0.916 0.996 1.100
gamma0_hat 100 0 1.012 0.291 0.075 0.997 1.996
gamma1_hat 100 0 2.384 16.937 −40.210 2.903 41.254
gamma2_hat 100 0 0.997 0.044 0.879 1.000 1.134
gamma3_hat 100 0 1.004 0.048 0.911 1.003 1.147
theta_hat 100 0 0.494 0.096 0.264 0.503 0.706
theta_hat_t_stats 100 0 5.299 0.847 3.097 5.434 7.016
R2_demand 100 0 0.356 0.048 0.230 0.362 0.464
R2_supply 100 0 0.974 0.002 0.968 0.973 0.978
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.007 0.089 9.789 10.003 10.203
alpha1_hat 100 0 0.465 6.336 −19.340 0.529 15.995
alpha2_hat 100 0 20.081 0.789 18.157 20.126 22.977
alpha3_hat 100 0 1.002 0.019 0.961 1.000 1.045
gamma0_hat 100 0 1.005 0.134 0.709 1.012 1.319
gamma1_hat 100 0 0.452 7.439 −24.174 1.048 16.892
gamma2_hat 100 0 1.000 0.020 0.952 1.000 1.051
gamma3_hat 100 0 0.999 0.021 0.948 0.998 1.053
theta_hat 100 0 0.501 0.045 0.426 0.496 0.628
theta_hat_t_stats 100 0 12.258 0.859 10.663 12.256 14.252
R2_demand 100 0 0.356 0.021 0.296 0.357 0.399
R2_supply 100 0 0.974 0.001 0.970 0.974 0.976

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